Multiple Correct Answers MCQs for Sub-Topics of Topic 5: Construction

Question 1. Which instrument is primarily used to construct a circle with a given radius?

(A) Ruler

(B) Protractor

(C) Compass

(D) Set square

Question 2. If the diameter of a circle is given as $14$ cm, what should be the compass setting (radius) to construct this circle?

(A) $14$ cm

(B) $28$ cm

(C) $7$ cm

(D) $3.5$ cm

Question 3. To construct a line segment of length $6.5$ cm using a ruler, where should you ideally place the starting point of the ruler's scale?

(A) At the $0$ mark

(B) At the $1$ mm mark

(C) At any convenient mark, then subtract

(D) At the $1$ cm mark

Question 4. To copy a line segment AB using a compass and ruler, what is the first step?

(A) Draw a line L longer than AB.

(B) Set the compass width to the length of AB.

(C) Mark a point P on line L.

(D) Draw an arc from point P.

Question 5. When constructing a circle with a compass, the stationary point of the compass indicates the:

(A) Radius

(B) Diameter

(C) Centre

(D) Arc

Question 6. If you want to construct a line segment exactly $8$ cm long, which instrument guarantees the most accurate length measurement in standard geometry tools?

(A) A ruler marked in centimetres

(B) A divider

(C) A protractor

(D) A set square

Question 7. To copy a line segment of length 'l', you set the compass to length 'l'. If you then mark two points on a line using this compass setting, the distance between them will be:

(A) Less than l

(B) Equal to l

(C) Greater than l

(D) Twice l

Question 8. Which of the following statements is true about constructing a circle with a given diameter?

(A) You set the compass to the diameter length.

(B) You set the compass to half the diameter length.

(C) You set the compass to twice the diameter length.

(D) The diameter is irrelevant for compass setting.

Question 9. When constructing a line segment of a specific length using a ruler, it is advisable to start measuring from the '0' mark because:

(A) It is the international standard.

(B) The edge of the ruler might be worn or damaged, while the '0' mark is usually more precise.

(C) It makes calculations easier (you read the single endpoint value).

(D) All rulers are manufactured perfectly from the edge.

Question 10. What is the primary purpose of copying a line segment?

(A) To measure its length accurately.

(B) To transfer its length to another location in the drawing.

(C) To find its midpoint.

(D) To draw a line parallel to it.

Question 11. If you are given a radius of $5$ cm and a specific point as the centre, how many unique circles can you construct with this radius and centre?

(A) Exactly one

(B) Two

(C) A finite number

(D) An infinite number

Question 12. When copying a line segment AB onto a line L starting from point P, you place the compass point at P and draw an arc intersecting L at Q. What does the segment PQ represent?

(A) The centre of a circle

(B) The midpoint of AB

(C) The copied line segment

(D) A random segment on L

Question 13. What is the minimum number of distinct points required to define a unique line segment?

(A) One

(B) Two

(C) Three

(D) Infinite

Question 14. When constructing a circle, the distance from the fixed central point to any point on the curve is called the:

(A) Diameter

(B) Chord

(C) Radius

(D) Arc length

Question 15. If you need to construct a line segment of length $x$, using a ruler marked in millimetres, what is the smallest unit you can typically measure and mark accurately?

(A) $1$ cm

(B) $1$ mm

(C) $0.5$ cm

(D) $0.1$ mm

Question 16. Which of the following cannot be used *alone* to construct a line segment of a specific given length?

(A) Ruler

(B) Compass

(C) Protractor

(D) Set square

Question 17. To construct a circle with diameter $D$, the compass opening should be:

(A) $D$

(B) $2D$

(C) $D/2$

(D) $\sqrt{D}$

Question 18. When copying a line segment, which fundamental property are you preserving?

(A) Orientation

(B) Position

(C) Length

(D) Colour

Question 19. A point is a basic geometric element that defines:

(A) Length only

(B) Area only

(C) Position only

(D) Volume only

Question 20. A line segment is a finite part of a line that has:

(A) One endpoint

(B) Two endpoints

(C) Infinite length

(D) No endpoints

Question 21. Which instrument is essential for drawing a straight line segment when given two points?

(A) Compass

(B) Protractor

(C) Ruler (or straight edge)

(D) Divider

Question 22. To construct a circle, you must know:

(A) The radius and the position of the centre.

(B) The diameter and the position of the centre.

(C) The circumference.

(D) The area.

Question 23. Copying a line segment is a fundamental technique useful for:

(A) Finding the slope of the segment.

(B) Creating parallel lines at a specific distance.

(C) Constructing other geometric figures that require precise lengths based on existing segments.

(D) Measuring the segment in different units.

Question 24. If you construct a circle with a radius of $4$ cm, what is its diameter?

(A) $2$ cm

(B) $4$ cm

(C) $8$ cm

(D) $16$ cm

Question 25. Which tools are used in conjunction to measure or transfer the distance between two points?

(A) Protractor

(B) Compass and ruler

(C) Set square

(D) Only compass

Question 26. Which of the following statements are true regarding the construction of a line segment of a specific length?

(A) A ruler with markings is typically used.

(B) A compass can be used to transfer a known length.

(C) The accuracy depends on the precision of the measuring tool and the marking instrument.

(D) A protractor is essential for ensuring the segment is straight.

Question 27. To construct a circle with a given radius, say R, which actions are correct?

(A) Open the compass so the distance between the point and the pencil is R.

(B) Choose a point to be the centre of the circle.

(C) Place the sharp point of the compass at the chosen centre.

(D) Rotate the compass while keeping the sharp point fixed, allowing the pencil to draw the curve.

Question 28. Copying a line segment AB onto a line L starting at P involves which key actions with a compass?

(A) Setting the compass opening to the length of AB.

(B) Placing the compass point at P.

(C) Drawing an arc that intersects line L.

(D) Drawing a full circle centered at P with radius AB.

Question 29. Which of the following correctly describes the properties of a line segment?

(A) It has a finite length.

(B) It is defined by two distinct endpoints.

(C) It is a subset of a line.

(D) It has no width or depth.

Question 30. If a circle has a radius of 'r', then its diameter 'd' and circumference 'C' are related by which formulas?

(A) $d = 2r$

(B) $C = \pi d$

(C) $C = 2 \pi r$

(D) $r = d/2$

Question 31. To construct a line segment of length L using a ruler and pencil, you will:

(A) Use the straight edge of the ruler to guide the pencil.

(B) Use the markings on the ruler to measure the length L.

(C) Mark two points separated by the distance L.

(D) Need to use a protractor to ensure the line is straight.

Question 1. To construct a $60^\circ$ angle using a compass and ruler, which steps are correct?

(A) Draw a ray with endpoint A.

(B) With A as centre, draw an arc of any radius intersecting the ray at B.

(C) With B as centre and the *same* radius, draw another arc intersecting the first arc at C.

(D) Join A to C. $\angle BAC = 60^\circ$.

Question 2. To construct a $90^\circ$ angle at a point P on a line, which methods are valid using compass and ruler?

(A) Construct the perpendicular bisector of a line segment centered at P.

(B) Draw an arc centered at P intersecting the line at A and B. Draw arcs from A and B with radius greater than AP, intersecting at Q. Join PQ.

(C) Construct a $60^\circ$ angle and add a $30^\circ$ angle adjacent to it.

(D) Bisect a straight angle (180 degrees) using the perpendicular bisector method.

Question 3. Which of the following angles can be constructed using only a compass and ruler through standard methods or simple combinations/bisections?

(A) $30^\circ$

(B) $45^\circ$

(C) $75^\circ$ ($60^\circ+15^\circ$)

(D) $105^\circ$ ($60^\circ+45^\circ$ or $90^\circ+15^\circ$)

Question 4. To bisect an angle $\angle PQR$, which steps are correct using compass and ruler?

(A) With Q as centre, draw an arc intersecting QP at A and QR at B.

(B) With A as centre, draw an arc in the interior of $\angle PQR$.

(C) With B as centre and the *same* radius as in (B), draw another arc intersecting the first arc at C.

(D) Join Q to C. QC is the angle bisector of $\angle PQR$.

Question 5. The justification for the angle bisector construction (QC bisecting $\angle PQR$) relies on congruence. If you connect A and B to C, and consider $\triangle QAC$ and $\triangle QBC$, which congruence criterion and corresponding parts (CPCT) are used?

(A) QA = QB (radii of the same arc).

(B) AC = BC (radii of equal arcs).

(C) QC is a common side.

(D) $\triangle QAC \cong \triangle QBC$ by SSS. Therefore, $\angle AQC = \angle BQC$ (CPCT), meaning QC bisects $\angle PQR$.

Question 6. To construct a $120^\circ$ angle at point O on a ray OA, you can:

(A) Construct a $60^\circ$ angle and then construct another $60^\circ$ angle adjacent to it on the same side, forming a larger angle.

(B) Construct a $60^\circ$ angle on the opposite side of OA on a line through O, then the angle supplementary to it on the same side of the line as OA is $120^\circ$.

(C) Construct a perpendicular at O, then add $30^\circ$ and $60^\circ$ to it.

(D) By extending the $60^\circ$ ray through A to form a straight line, the adjacent angle is $120^\circ$. Construct the $60^\circ$ angle and then use the straight line.

Question 7. Which angles can be derived by repeatedly bisecting a $60^\circ$ angle?

(A) $30^\circ$

(B) $15^\circ$

(C) $7.5^\circ$

(D) $22.5^\circ$

Question 8. The construction of an angle bisector divides the angle into two angles of equal measure. This implies:

(A) The two resulting angles are congruent.

(B) Any point on the bisector is equidistant from the vertex of the angle.

(C) Any point on the bisector is equidistant from the two arms of the angle.

(D) The sum of the two angles formed by the bisector is equal to the original angle.

Question 9. To construct a $45^\circ$ angle, you first construct a $90^\circ$ angle and then:

(A) Bisect one of the arms of the $90^\circ$ angle.

(B) Bisect the $90^\circ$ angle.

(C) Construct another $45^\circ$ angle adjacent to it.

(D) Subtract $45^\circ$ from the $90^\circ$ angle (in terms of construction steps).

Question 10. Which of the following require the construction of an angle bisector as an intermediate step?

(A) Constructing a $30^\circ$ angle (by bisecting $60^\circ$).

(B) Constructing a $45^\circ$ angle (by bisecting $90^\circ$).

(C) Constructing a $15^\circ$ angle (by bisecting $30^\circ$).

(D) Constructing the incenter of a triangle (intersection of angle bisectors).

Question 11. In the construction of angles like $60^\circ$, the radius used for drawing arcs from the vertex and then from the intersection point on the ray must be the same. Why is this important?

(A) To ensure the two arcs intersect.

(B) To create an equilateral triangle within the construction, guaranteeing the $60^\circ$ angle.

(C) To define a specific point that is equidistant from A and B (where A is the vertex, B is on the ray).

(D) It is just a convention and any radius works.

Question 12. A point on the angle bisector of $\angle XYZ$ is $7$ cm away from ray YX. What is its distance from ray YZ?

(A) Less than $7$ cm

(B) Exactly $7$ cm

(C) More than $7$ cm

(D) Cannot be determined from the information given.

Question 13. To construct a $135^\circ$ angle, you can combine standard constructions. Which combinations are valid?

(A) Construct $90^\circ$ and add $45^\circ$ adjacent to it.

(B) Construct a $180^\circ$ straight angle and subtract $45^\circ$ (construct $45^\circ$ on the line such that the remaining angle is $135^\circ$).

(C) Bisect a $270^\circ$ reflex angle.

(D) Add $60^\circ, 60^\circ,$ and $15^\circ$.

Question 14. What is the result of bisecting a right angle ($90^\circ$)?

(A) An acute angle

(B) An obtuse angle

(C) Two angles of $45^\circ$ each

(D) A straight angle

Question 15. Which tools are necessary and sufficient for angle bisection in traditional Euclidean geometry?

(A) Compass

(B) Ruler (straight edge is needed to draw the bisector ray)

(C) Protractor

(D) Set square

Question 16. When constructing a $60^\circ$ angle, the points A, B, and C (vertex, point on ray, intersection of arcs) form an equilateral triangle. This is why the angle is $60^\circ$.

(A) True

(B) False

(C) Only if the initial radius is $1$ unit.

(D) This forms an isosceles triangle, not necessarily equilateral.

Question 17. Which of the following is NOT a standard angle that can be constructed directly or by simple bisections/combinations of $60^\circ$ and $90^\circ$ with compass and ruler?

(A) $7.5^\circ$

(B) $10^\circ$

(C) $15^\circ$

(D) $22.5^\circ$

Question 18. The ray that divides an angle into two angles of equal measure is called the:

(A) Median

(B) Altitude

(C) Perpendicular bisector

(D) Angle bisector

Question 19. The construction of a $15^\circ$ angle can be achieved by:

(A) Bisecting a $30^\circ$ angle.

(B) Bisecting the angle between the $60^\circ$ mark and the $90^\circ$ mark on an arc from the vertex (which is $30^\circ$).

(C) Subtracting $45^\circ$ from $60^\circ$.

(D) Bisecting a $7.5^\circ$ angle.

Question 20. Which of the following angles can be constructed by first constructing a $90^\circ$ angle and then performing bisections?

(A) $45^\circ$

(B) $22.5^\circ$

(C) $11.25^\circ$

(D) $30^\circ$

Question 1. To construct a perpendicular to a line 'l' at a point P on 'l', which steps are typically involved?

(A) With P as center, draw arcs of equal radius intersecting 'l' at A and B.

(B) With A as center, draw an arc above or below 'l'.

(C) With B as center and the *same* radius as the arc from A, draw another arc intersecting the first arc at Q.

(D) Join P to Q. The line PQ is perpendicular to 'l' at P.

Question 2. To construct a perpendicular to a line 'l' from a point P *outside* 'l', which steps are typically involved?

(A) With P as center, draw an arc intersecting 'l' at two points, say A and B.

(B) With A as center, draw an arc on the side of 'l' opposite to P.

(C) With B as center and the *same* radius as the arc from A, draw another arc intersecting the first arc at Q.

(D) Join P to Q. The line PQ is perpendicular to 'l'.

Question 3. Which of the following are properties of the perpendicular bisector of a line segment AB?

(A) It forms a $90^\circ$ angle with AB.

(B) It passes through the midpoint of AB.

(C) Any point on the bisector is equidistant from A and B.

(D) Every point equidistant from A and B lies on the perpendicular bisector.

Question 4. To construct the perpendicular bisector of a line segment AB, which actions are correct?

(A) With A as centre, draw an arc above and below AB with a radius greater than half the length of AB.

(B) With B as centre, draw arcs with the *same* radius as in (A) above and below AB, intersecting the previous arcs at P and Q.

(C) Join P to Q. The line PQ is the perpendicular bisector of AB.

(D) The radius used must be exactly equal to the length of AB.

Question 5. The construction of a perpendicular at a point on a line is essentially the construction of a $90^\circ$ angle. This method is equivalent to:

(A) Bisecting a straight angle ($180^\circ$).

(B) Applying the perpendicular bisector construction to a segment centered at the point on the line.

(C) Constructing an angle supplementary to a $90^\circ$ angle.

(D) Drawing a line that forms four $90^\circ$ angles with the original line.

Question 6. The justification for the perpendicular bisector construction relies on the fact that P and Q (the intersection points of the arcs) are equidistant from A and B. Thus, P and Q lie on the perpendicular bisector. Since two points define a unique line, the line PQ is the perpendicular bisector. Which congruence criteria are relevant here when considering triangles formed? (e.g., $\triangle APQ$ and $\triangle BPQ$; or $\triangle APC$ and $\triangle BPC$ where C is the intersection with AB)

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 7. When constructing a perpendicular from a point P outside a line, the initial arc from P intersects the line at A and B. This creates an isosceles triangle PAB where PA = PB. The perpendicular from P to the line is the altitude from P to the base AB. In an isosceles triangle, the altitude to the base is also the median and angle bisector of the vertex angle. Which property is primarily used?

(A) In an isosceles triangle, the altitude to the base is also the median.

(B) The shortest distance from a point to a line is along the perpendicular.

(C) All angles in a triangle sum to $180^\circ$.

(D) Congruence of triangles (e.g., $\triangle PAC \cong \triangle PBC$ where C is on AB).

Question 8. To find the midpoint of a line segment AB, you can:

(A) Measure the length of AB with a ruler and divide by 2, then mark the point at that distance from one endpoint.

(B) Construct the perpendicular bisector of AB; the point where it intersects AB is the midpoint.

(C) Use a compass to divide the segment into two equal parts by trial and error.

(D) Draw arcs of equal radius (greater than half AB) from A and B, intersecting at two points, then join these points.

Question 9. The construction of a perpendicular bisector is a key step in constructing which circle related to a triangle?

(A) Incircle

(B) Circumcircle (The circumcenter, equidistant from vertices, is the intersection of perpendicular bisectors of sides).

(C) Excircles

(D) Director circle

Question 10. Which statements are true about constructing a line perpendicular to a given line 'l' through a given point?

(A) If the point is on the line, there is a unique perpendicular line through it.

(B) If the point is outside the line, there is a unique perpendicular line through it.

(C) Perpendicular lines intersect at an angle of $90^\circ$.

(D) The constructed line is also a bisector if it passes through the midpoint of a segment on the original line.

Question 11. When constructing a perpendicular bisector of segment AB, why must the radius of the arcs drawn from A and B be greater than half the length of AB?

(A) To ensure the arcs intersect at two distinct points.

(B) If the radius is less than or equal to half AB, the arcs will not intersect (or will only touch at the midpoint).

(C) To create a rhombus formed by A, B, and the two intersection points, whose diagonals are perpendicular bisectors of each other.

(D) To make the constructed line longer.

Question 12. Which tools are necessary and sufficient for constructing a perpendicular bisector of a line segment in traditional Euclidean geometry?

(A) Compass

(B) Ruler (straight edge needed to draw the bisector line)

(C) Protractor

(D) Set square

Question 1. To construct a line parallel to a given line 'l' through a point P not on 'l', using corresponding angles, which steps are valid?

(A) Draw a transversal line through P intersecting 'l' at Q.

(B) Copy the angle formed by the transversal and 'l' at Q (a corresponding angle).

(C) Construct the copied angle at P in the corresponding position relative to the transversal and P.

(D) Draw a line through P and the point that defines the copied angle's arm.

Question 2. To construct a line parallel to a given line 'l' through a point P not on 'l', using alternate interior angles, which steps are valid?

(A) Draw a transversal line through P intersecting 'l' at Q.

(B) Copy the angle formed by the transversal and 'l' at Q (an alternate interior angle).

(C) Construct the copied angle at P in the alternate interior position relative to the transversal and P.

(D) The line forming the new angle at P is parallel to 'l'.

Question 3. The construction of parallel lines relies on the properties of angles formed by a transversal intersecting two other lines. Which angle relationships, if true, prove that the two lines are parallel?

(A) A pair of corresponding angles are equal.

(B) A pair of alternate interior angles are equal.

(C) A pair of interior angles on the same side of the transversal are supplementary (sum to $180^\circ$).

(D) A pair of vertically opposite angles are equal.

Question 4. To copy an angle using compass and ruler, which steps are involved?

(A) Draw an arc from the vertex of the original angle intersecting both arms.

(B) Draw a similar arc from the desired vertex of the new angle with the *same* radius.

(C) Measure the straight-line distance between the intersection points on the arc of the original angle using the compass.

(D) Transfer this distance as an arc onto the arc of the new angle using the compass, locating a point which defines the second arm.

Question 5. Which statement(s) are true about constructing a line parallel to a given line 'l' through a point P not on 'l'?

(A) There exists exactly one such line according to Euclidean geometry (Parallel Postulate).

(B) The construction is possible using only compass and ruler.

(C) The construction methods typically involve copying an angle precisely.

(D) The constructed line will intersect the original line at the external point P.

Question 6. In the corresponding angles method for parallel lines, after drawing the transversal through P and copying the angle at Q, you draw a line through P and the newly located point. Which geometric principle justifies that this new line is parallel to the original line 'l'?

(A) Basic Proportionality Theorem

(B) The converse of the Corresponding Angles Postulate (If corresponding angles are equal, lines are parallel).

(C) Angle Sum Property

(D) Corresponding angles formed by the transversal are equal.

Question 7. Parallel lines are defined as two lines in the same plane that:

(A) Have no point in common (never intersect).

(B) Maintain a constant distance between them.

(C) Form a $90^\circ$ angle with a perpendicular transversal.

(D) Have the same direction.

Question 8. The alternate interior angles method for constructing parallel lines works because:

(A) Alternate interior angles formed by a transversal are equal when the lines are parallel.

(B) The converse of the Alternate Interior Angles Theorem states that if alternate interior angles formed by a transversal are equal, then the lines are parallel.

(C) The construction ensures these angles are equal by copying one onto the other.

(D) Corresponding angles are related to alternate interior angles.

Question 9. When constructing a parallel line using the corresponding angles method, the transversal should ideally make an angle with the original line that is acute or obtuse (not $0^\circ$ or $180^\circ$). Why?

(A) To create distinct intersection points on the arc for copying the angle.

(B) To ensure the transversal intersects the line 'l'.

(C) To make the corresponding angles easily identifiable and constructible.

(D) A straight transversal would not create angles to copy.

Question 10. Which of the following are used as a basis for constructing parallel lines through an external point using compass and ruler?

(A) Copying corresponding angles.

(B) Copying alternate interior angles.

(C) Using the property that two lines perpendicular to the same line are parallel.

(D) Bisecting the angle between the external point and the line.

Question 1. To divide a line segment AB internally in the ratio m:n, using the standard construction method involving a ray AC, which steps are correct?

(A) Draw a ray AC making an acute angle with AB.

(B) Mark $m+n$ points $A_1, A_2, \dots, A_{m+n}$ on AC such that $AA_1 = A_1A_2 = \dots = A_{m+n-1}A_{m+n}$ (equal distances marked using a compass).

(C) Join $A_{m+n}$ to B.

(D) Through $A_m$, draw a line parallel to $A_{m+n}B$ intersecting AB at P. P is the required point such that AP:PB = m:n.

Question 2. When dividing a line segment AB in the ratio 5:2 internally, what is true about the construction?

(A) You draw a ray AC making an acute angle with AB.

(B) You mark 7 equal points on ray AC.

(C) You join the 7th point ($A_7$) to B.

(D) You draw a line through the 5th point ($A_5$) parallel to $A_7B$.

Question 3. The justification for dividing a line segment in a given ratio relies on the Basic Proportionality Theorem (BPT). This theorem applies to the triangle formed by AB and the segment $A_{m+n}B$. Which statements about the application of BPT are correct?

(A) In $\triangle ABA_{m+n}$, the line segment $PA_m$ is drawn parallel to $BA_{m+n}$.

(B) By BPT, $\frac{AP}{PB} = \frac{AA_m}{A_m A_{m+n}}$.

(C) Since $AA_m$ consists of 'm' equal parts and $A_m A_{m+n}$ consists of 'n' equal parts, $\frac{AA_m}{A_m A_{m+n}} = \frac{m}{n}$.

(D) Therefore, $\frac{AP}{PB} = \frac{m}{n}$.

Question 4. To divide a line segment of length 18 cm into 3 equal parts, using the construction method for division in a given ratio, which steps are valid?

(A) This is equivalent to dividing in the ratio 1:1:1, or finding points that divide it in ratios 1:2 and 2:1.

(B) You can divide the segment in ratio 1:2 at point P, and then divide PB in ratio 1:1 at point Q.

(C) You can mark 3 equal parts on the auxiliary ray and connect the 3rd point to one end of the segment, then draw parallel lines through the 1st and 2nd points.

(D) The total number of equal parts marked on the auxiliary ray should be 3.

Question 5. If a line segment AB is divided internally at point P in the ratio m:n, which of the following are true?

(A) P is a point on the line segment AB.

(B) The ratio of the length of AP to the length of PB is m/n.

(C) The sum of the lengths of AP and PB is equal to the length of AB.

(D) If m=n, P is the midpoint of AB.

Question 6. The construction for dividing a line segment in a given ratio requires the ability to perform which other basic constructions using compass and ruler?

(A) Drawing a line segment of any desired length.

(B) Copying a line segment (to mark equal parts on the ray).

(C) Constructing parallel lines.

(D) Constructing angles.

Question 7. Consider dividing a line segment AB in the ratio 3:1. On the ray AC, you mark 4 equal points $A_1, \dots, A_4$. You join $A_4$ to B. Which point on AC should you draw a parallel line from to intersect AB at the desired point P?

(A) $A_3$ (since m=3)

(B) $A_1$ (since n=1)

(C) $A_{3+1} = A_4$

(D) The point corresponding to the first part of the ratio.

Question 8. The justification for the line segment division construction demonstrates that the point P divides AB in the ratio m:n. This is because the parallel line through $A_m$ creates similar triangles, and the ratio of corresponding sides is equal to the ratio of the segments on the auxiliary ray, which was set up as m:n.

(A) True, the justification involves similar triangles and proportionality.

(B) False, congruence of triangles is used.

(C) The ratio of segments on AB is equal to the ratio of segments on AC by BPT.

(D) The construction ensures that the line segment is accurately measured before division.

Question 9. If you divide a line segment of length L in the ratio 1:1 internally, which of the following are true?

(A) The division point is the midpoint of the segment.

(B) The segment is divided into two parts of length L/2.

(C) This is the only ratio that results in equal parts.

(D) This requires marking $1+1=2$ equal parts on the auxiliary ray.

Question 10. Which statements about the ray AC used in the construction for dividing a line segment AB in ratio m:n are correct?

(A) It must originate from point A.

(B) It must not lie on the line containing AB.

(C) It must make an acute angle with AB (though any angle other than $0^\circ$ or $180^\circ$ works, acute is standard practice).

(D) The length of the equal segments marked on it influences the division ratio m:n.

Question 11. When dividing a segment AB in ratio m:n, you draw ray AC. Why is it specified that C is not on the line AB?

(A) To prevent the auxiliary ray from overlapping with the segment AB.

(B) To create a triangle $ABA_{m+n}$ (or similar setup) necessary for applying the BPT.

(C) To ensure the equal divisions on AC are clearly distinct from AB.

(D) It is just a visual preference.

Question 1. To construct a triangle given the lengths of its three sides a, b, and c (SSS Criterion), which steps are correct?

(A) Draw one side, say length 'a', as the base.

(B) From one endpoint of the base, draw an arc with radius 'b'.

(C) From the other endpoint of the base, draw an arc with radius 'c'.

(D) The intersection of the arcs gives the third vertex of the triangle.

Question 2. A triangle can be uniquely constructed (up to congruence) if you are given:

(A) The lengths of three sides (provided the Triangle Inequality holds).

(B) The lengths of two sides and the measure of the included angle (SAS).

(C) The measures of two angles and the length of the included side (ASA).

(D) The measures of two angles and the length of a non-included side (AAS).

Question 3. To construct a triangle given two sides (say 7 cm and 9 cm) and the included angle (say $70^\circ$), which steps are valid?

(A) Draw a line segment of length 7 cm.

(B) At one endpoint of the 7 cm segment, construct an angle of $70^\circ$.

(C) Along the arm of the $70^\circ$ angle, measure and mark a segment of length 9 cm.

(D) Join the endpoint of the 9 cm segment to the other endpoint of the 7 cm segment.

Question 4. In constructing a triangle using the ASA criterion (Two Angles and Included Side), if the given angles are $\alpha$ and $\beta$ and the included side is 's', which statement(s) are true?

(A) The sum of the two given angles must be less than $180^\circ$ ($ \alpha + \beta < 180^\circ$).

(B) You draw the side 's' as the base.

(C) You construct angle $\alpha$ at one endpoint of 's' and angle $\beta$ at the other endpoint, both on the same side of 's'.

(D) The intersection of the arms of angles $\alpha$ and $\beta$ forms the third vertex of the triangle.

Question 5. If you are given two angles and one side of a triangle (AAS Criterion), and the side is NOT included between the given angles, you can still construct the triangle. Which statements explain how?

(A) Use the angle sum property ($180^\circ$) to find the measure of the third angle.

(B) Since you now have all three angles and one side, the problem is reduced to an ASA construction because the given side will be included between one of the given angles and the newly calculated angle.

(C) The problem is reduced to an SAS construction.

(D) The construction is impossible with compass and ruler alone unless the side is included.

Question 6. Which of the following sets of side lengths can form a triangle?

(A) 4 cm, 5 cm, 6 cm

(B) 3 cm, 4 cm, 7 cm (3+4=7, fails strict triangle inequality)

(C) 6 cm, 6 cm, 6 cm

(D) 2 cm, 5 cm, 8 cm (2+5=7 < 8, fails triangle inequality)

Question 7. In the SSS construction, if the arcs drawn from the endpoints of the base with radii equal to the other two sides intersect at two points (one above and one below the base), which statements are true?

(A) Both intersection points define a valid third vertex.

(B) The two triangles formed are congruent reflections of each other across the base.

(C) You can choose either point to form the triangle.

(D) The triangle inequality theorem holds for the given side lengths.

Question 8. The construction of a triangle based on given measurements often requires accurate measurement and transfer of lengths and angles. Which tools are primarily used for these tasks in traditional compass and ruler constructions?

(A) Ruler (for measuring/drawing line segments of specified length)

(B) Compass (for transferring lengths and drawing arcs to find intersection points)

(C) Protractor (generally avoided in pure compass and ruler constructions, used for measuring/drawing angles otherwise)

(D) Divider (can be used for transferring lengths)

Question 9. If you are given two sides and a non-included angle (SSA case), a unique triangle is NOT always determined. Which statements are true about this case?

(A) It is known as the ambiguous case of triangle construction.

(B) It is possible to construct two different triangles with the same given measurements in certain situations.

(C) If the given angle is obtuse, the triangle is uniquely determined.

(D) If the side opposite the given angle is longer than the other given side, the triangle is uniquely determined.

Question 10. Which combinations of three measurements are sufficient to uniquely construct a triangle?

(A) Three angles (determines shape, but not size).

(B) Two sides and the included angle (SAS).

(C) Two angles and the included side (ASA).

(D) Two sides and the angle opposite to one of them (SSA) where the angle is $90^\circ$ (RHS).

Question 1. To construct an equilateral triangle with side length 'a', which methods are valid?

(A) Draw a segment of length 'a', then from each endpoint draw an arc of radius 'a'. Their intersection is the third vertex.

(B) Draw a segment of length 'a', then construct $60^\circ$ angles at both endpoints. The intersection of the arms forms the third vertex.

(C) Use the SSS criterion with all three sides equal to 'a'.

(D) Use the ASA criterion with one side 'a' and two adjacent angles of $60^\circ$.

Question 2. The construction of an equilateral triangle by drawing arcs of radius equal to the side length from the endpoints of the base is justified because:

(A) The three sides of the resulting triangle are guaranteed to be equal in length by the construction process.

(B) A triangle with three equal sides is, by definition, equilateral.

(C) This construction relies on the property that in an equilateral triangle, all sides are equal.

(D) It forms an isosceles triangle on the base with the third vertex, where the two equal sides are the radii used.

Question 3. To construct an isosceles triangle given the base length and the length of the two equal sides, which construction method applies?

(A) SSS criterion.

(B) You draw the base and then use the length of the equal sides as radii for arcs from the base endpoints to find the third vertex.

(C) SAS criterion (if the vertex angle is also given, along with the equal sides).

(D) ASA criterion (if the base and base angles are given).

Question 4. To construct an isosceles triangle given the base length and the base angles, which criterion applies and which steps are valid?

(A) ASA criterion.

(B) You draw the base and construct the given base angles at its endpoints.

(C) The angles constructed at the base must be equal for it to be isosceles with those as base angles.

(D) The intersection of the arms of the base angles forms the third vertex.

Question 5. To construct a right-angled triangle given the hypotenuse and one leg (RHS criterion), which steps are correct?

(A) Draw the given leg as one side (say AB).

(B) At one endpoint of the leg (say A), construct a $90^\circ$ angle.

(C) From the other endpoint of the leg (B), draw an arc with radius equal to the hypotenuse length.

(D) The intersection of the arc and the arm of the $90^\circ$ angle is the third vertex (C).

Question 6. The justification for the RHS construction relies on the properties of right-angled triangles. Which properties are relevant?

(A) The Pythagorean theorem ($a^2 + b^2 = c^2$) applies to the sides of the resulting triangle, confirming it's right-angled with the correct side lengths.

(B) The angle sum property ($A+B+C=180^\circ$) applies.

(C) The RHS congruence criterion for triangles confirms that if two right triangles have congruent hypotenuses and one congruent leg, they are congruent.

(D) This construction method ensures that one angle is exactly $90^\circ$ and the side opposite the right angle is the hypotenuse of the given length.

Question 7. An isosceles triangle has at least two equal sides. Which properties are always true for an isosceles triangle?

(A) The angles opposite the equal sides are equal (base angles).

(B) It has at least one axis of symmetry.

(C) The angle bisector of the vertex angle is also the median and altitude to the base.

(D) All three angles are always equal.

Question 8. When constructing a right-angled triangle using the RHS criterion, it is crucial that the hypotenuse length is strictly greater than the given leg length. Why?

(A) The hypotenuse is the longest side in any right-angled triangle.

(B) Geometrically, if the hypotenuse length equals the leg length, the arc would only touch the $90^\circ$ arm at the base endpoint.

(C) If the hypotenuse length is less than the leg length, the arc will not intersect the $90^\circ$ arm away from the base.

(D) $c^2 = a^2 + b^2$. If $c \le a$, then $c^2 \le a^2$, implying $b^2 \le 0$, which means no real side length 'b' exists for the other leg (unless b=0 for $c=a$).

Question 9. Which statements are true about equilateral triangles?

(A) All three sides are equal in length.

(B) All three interior angles are equal to $60^\circ$.

(C) Every equilateral triangle is a special case of an isosceles triangle.

(D) The circumcenter, incenter, centroid, and orthocenter all coincide at a single point.

Question 10. To construct an isosceles triangle given the base and the vertex angle, which steps are necessary?

(A) Draw the base AB.

(B) Calculate the measure of the base angles using the angle sum property: Base angle = $(180^\circ - \text{vertex angle})/2$.

(C) Construct angles equal to the calculated base angle at points A and B on the same side of AB.

(D) The intersection of the arms of these angles forms the third vertex C.

Question 11. When constructing a right-angled triangle using the RHS criterion, one angle is already fixed at $90^\circ$. What is true about the other two angles?

(A) They must be acute.

(B) Their sum is $90^\circ$ (complementary).

(C) They can be equal (for an isosceles right triangle).

(D) One angle can be obtuse if the other is acute.

Question 12. Which tools are necessary and sufficient for constructing basic triangles (SSS, SAS, ASA, RHS) in traditional Euclidean geometry?

(A) Compass

(B) Ruler (straight edge, plus markings for given lengths)

(C) Protractor (avoided in pure compass/ruler constructions)

(D) Pencil/Pen

Question 1. To construct a triangle ABC given base BC, angle $\angle B$, and the sum of the other two sides (AB + AC), which steps are part of the common construction method?

(A) Draw BC as the base.

(B) Construct $\angle B$ at point B.

(C) On the ray from B forming $\angle B$, mark a point D such that BD = AB + AC.

(D) Join D to C and construct the perpendicular bisector of CD.

Question 2. In the construction of $\triangle ABC$ given BC, $\angle B$, and AB + AC, the perpendicular bisector of CD intersects the ray BD at point A. Which geometric properties justify that this is the correct position for vertex A?

(A) Any point on the perpendicular bisector of CD is equidistant from C and D, so AC = AD.

(B) By construction, BD = AB + AC.

(C) Point A lies on BD, so BD = BA + AD.

(D) Substituting AC = AD into BD = BA + AD gives BD = BA + AC, which matches the given condition.

Question 3. To construct a triangle ABC given base BC, angle $\angle B$, and the difference of the other two sides (|AB - AC|), which steps are part of the common construction method (assuming AB > AC)?

(A) Draw BC as the base.

(B) Construct $\angle B$ at point B.

(C) On the ray from B forming $\angle B$, mark a point D such that BD = AB - AC.

(D) Join D to C and construct the perpendicular bisector of CD.

Question 4. In the construction of $\triangle ABC$ given BC, $\angle B$, and AB - AC, the perpendicular bisector of CD intersects the ray BX at A. This method is valid because A being on the perpendicular bisector of CD implies AC = AD. Given AB > AC, we have AB = AD + DB. Substituting AD = AC, we get AB = AC + DB, so AB - AC = DB. Which statements are true?

(A) A is the intersection of the perpendicular bisector of CD and the ray BX.

(B) Triangle ADC is an isosceles triangle with AC = AD.

(C) The length BD is equal to the difference of the other two sides (AB - AC).

(D) If AC > AB, point D is marked on the extension of the ray BX *backwards* from B, and AC = AD still holds.

Question 5. To construct a triangle ABC given angles $\angle B$ and $\angle C$ and the perimeter (AB + BC + CA), which steps are involved?

(A) Draw a line segment PQ equal to the perimeter.

(B) At P, construct an angle equal to $(1/2)\angle B$.

(C) At Q, construct an angle equal to $(1/2)\angle C$.

(D) The intersection of the arms of these angles is vertex A. Then construct perpendicular bisectors of AP and AQ. Their intersections with PQ give points B and C.

Question 6. In the perimeter construction for $\triangle ABC$ (given $\angle B$, $\angle C$, Perimeter), the justification for constructing angles $(1/2)\angle B$ and $(1/2)\angle C$ and using perpendicular bisectors of AP and AQ relies on creating isosceles triangles. Which statements explain this?

(A) By construction, $\triangle APB$ is isosceles with angles $\angle APB = \angle PAB = (1/2)\angle B$, so BP = BA.

(B) By construction, $\triangle AQC$ is isosceles with angles $\angle AQC = \angle QAC = (1/2)\angle C$, so CQ = CA.

(C) B lies on the perpendicular bisector of AP, so BP = BA.

(D) C lies on the perpendicular bisector of AQ, so CQ = CA.

Question 7. To construct a triangle given two sides (say AB, AC) and the median AD to BC, a common method involves extending AD to E such that AD = DE, and joining C to E. This creates parallelogram ABEC. Which properties are used?

(A) The diagonals of ABEC bisect each other at D.

(B) Opposite sides of ABEC are equal (AB = CE, AC = BE).

(C) Triangle ACE can be constructed using SSS criterion with sides AC, CE (=AB), and AE (=2*AD).

(D) Once $\triangle ACE$ is constructed, the midpoint D of AE is found, and extending CD to B such that CD = DB locates B.

Question 8. To construct a triangle given two angles ($\angle B, \angle C$) and the altitude AD to BC, which statements describe the initial and final steps?

(A) Draw a line and mark a point D on it.

(B) Construct a perpendicular to the line at D and mark A on it such that AD equals the given altitude length.

(C) Construct angles at A whose arms intersect the original line at B and C. These angles at A are related to angles B and C (e.g., $\angle DAB = 90^\circ - \angle B$).

(D) Construct angles equal to $\angle B$ and $\angle C$ directly at points on the perpendicular line through D.

Question 9. Advanced triangle constructions often involve transforming the problem into a simpler construction or utilizing properties of specific geometric figures. Which types of transformations or properties are commonly used?

(A) Using perpendicular bisectors to exploit the property of equidistance from two points.

(B) Creating isosceles triangles to relate side sums or differences to known lengths.

(C) Forming parallelograms to use properties of diagonals and opposite sides.

(D) Using angle bisectors to relate distances from the arms of an angle.

Question 10. In the construction of a triangle given one side, one angle, and the difference of the other two sides, if AC > AB, point D is marked on the extension of ray CB backwards from B such that BD = AC - AB. The perpendicular bisector of CD is constructed. Which statements are true?

(A) The perpendicular bisector intersects the ray BX (forming $\angle B$) at point A.

(B) Triangle ADC is isosceles with AC = AD.

(C) AB = AD - BD.

(D) AC - AB = BD, which matches the construction.

Question 1. To construct a triangle similar to a given triangle ABC with a scale factor $k = m/n$ (where m, n are positive integers) sharing vertex B and having BC on the same line as the new side BC', which initial steps are correct?

(A) Draw triangle ABC.

(B) Draw a ray BX from B making an acute angle with BC.

(C) Mark $\text{max}(m, n)$ points $B_1, B_2, \dots, B_{\text{max}(m,n)}$ on BX such that $BB_1 = A_1A_2 = \dots$ (equal distances marked using a compass).

(D) The total number of equal parts marked on BX is $m+n$.

Question 2. If the scale factor is $k = m/n$ and $m < n$ (scale down), you connect $B_n$ to C. To find C' on BC, you draw a line through $B_m$ parallel to $B_n$C. Which statements are true about the resulting segment BC'?

(A) $C'$ lies on the segment BC.

(B) The length $BC'$ is less than $BC$.

(C) $\frac{BC'}{BC} = \frac{m}{n}$ by BPT.

(D) $\triangle BB_m C'$ is congruent to $\triangle BB_n C$.

Question 3. If the scale factor is $k = m/n$ and $m > n$ (scale up), you connect $B_n$ to C. To find C' on the extension of BC, you draw a line through $B_m$ parallel to $B_n$C. Which statements are true about the resulting segment BC'?

(A) $C'$ lies on the extension of BC beyond C.

(B) The length $BC'$ is greater than $BC$.

(C) $\frac{BC'}{BC} = \frac{m}{n}$ by BPT.

(D) $\triangle BB_n C$ is similar to $\triangle BB_m C'$.

Question 4. The justification for the similar triangle construction relies on the concept of similarity. Which criteria or theorems are relevant in proving $\triangle A'BC' \sim \triangle ABC$?

(A) Basic Proportionality Theorem (BPT) is used to establish the ratio of sides on the base BC and the arm BA.

(B) AA (Angle-Angle) similarity criterion, as parallel lines A'C' and AC create equal corresponding angles ($\angle BA'C' = \angle BAC$, $\angle BC'A' = \angle BCA$).

(C) SAS (Side-Angle-Side) similarity criterion, as $\angle B$ is common and the sides forming the angle are proportional ($\frac{BA'}{BA} = \frac{BC'}{BC} = k$).

(D) SSS (Side-Side-Side) similarity criterion, as all corresponding sides are shown to be in the ratio k.

Question 5. To complete the similar triangle $A'BC'$ after finding $C'$ on BC (or its extension), you draw a line through $C'$ parallel to AC. This line intersects BA (or its extension) at $A'$. Which statements are true about the resulting vertex $A'$?

(A) $A'$ lies on the line containing BA.

(B) If $k < 1$, $A'$ lies on the segment BA.

(C) If $k > 1$, $A'$ lies on the extension of BA beyond A.

(D) The line segment $A'C'$ is parallel to AC.

Question 6. If the scale factor for constructing a similar triangle is $4/3$, which statements are true?

(A) The new triangle will be larger than the original.

(B) You mark 4 points on the auxiliary ray.

(C) You connect $B_3$ to C and draw a parallel line through $B_4$.

(D) The vertices of the new triangle will lie on the extensions of the sides of the original triangle (away from B).

Question 7. If the scale factor for constructing a similar triangle is $2/5$, which statements are true?

(A) The new triangle will be smaller than the original.

(B) You mark 5 points on the auxiliary ray.

(C) You connect $B_5$ to C and draw a parallel line through $B_2$.

(D) The vertices of the new triangle will lie on the interior of the sides of the original triangle (starting from B).

Question 8. The construction of similar triangles relies on accurately drawing parallel lines. Which methods can be used to draw the necessary parallel lines in this construction?

(A) Using a compass and ruler to copy corresponding angles.

(B) Using a compass and ruler to copy alternate interior angles.

(C) Using set squares and a ruler (as a practical method).

(D) By constructing perpendiculars to a transversal at specific points.

Question 9. If $\triangle A'BC'$ is similar to $\triangle ABC$ with a scale factor $k = m/n$, which of the following relationships hold?

(A) The ratio of corresponding side lengths is k.

(B) The ratio of their perimeters is k.

(C) The ratio of their corresponding angle measures is k.

(D) The ratio of their areas is $k^2$.

Question 10. When constructing a similar triangle with a scale factor of 1, the constructed triangle will be:

(A) Congruent to the original triangle.

(B) An exact copy of the original triangle (in terms of shape and size).

(C) Have side lengths equal to the original triangle.

(D) Have angle measures equal to the original triangle.

Question 1. A unique quadrilateral can be constructed if you are given:

(A) The lengths of four sides and the measure of one angle.

(B) The lengths of three sides and the measures of two included angles.

(C) The lengths of two adjacent sides and the measures of three angles.

(D) The lengths of four sides and the length of one diagonal.

Question 2. To construct a general quadrilateral given four sides and a diagonal, say sides AB, BC, CD, DA and diagonal AC, which initial steps are correct?

(A) Draw the diagonal AC as the base for two triangles.

(B) Construct triangle ABC using SSS criterion with sides AB, BC, and AC.

(C) Construct triangle ADC using SSS criterion with sides AD, CD, and AC.

(D) The vertices B and D should be constructed on the same side of the diagonal AC.

Question 3. Which sets of information are sufficient for the unique construction of a parallelogram?

(A) Two adjacent sides and the included angle.

(B) Two adjacent sides and a diagonal.

(C) Both diagonals and the angle between them.

(D) Length of one side and the measures of two adjacent angles.

Question 4. To construct a rectangle given its length 'l' and width 'w', which methods are valid?

(A) Draw side 'l', construct $90^\circ$ angles at endpoints, mark 'w' on perpendiculars, join endpoints of 'w'.

(B) Draw side 'l', construct $90^\circ$ at one endpoint, mark 'w' on the perpendicular. From the other end of 'l', draw an arc of length 'w'. From the endpoint of 'w', draw an arc of length 'l'. Their intersection is the fourth vertex.

(C) Use the properties of a parallelogram with a $90^\circ$ angle.

(D) Use the properties of a square with unequal adjacent sides.

Question 5. A rhombus is a parallelogram with all four sides equal. Which properties are useful for constructing a rhombus?

(A) All sides are equal.

(B) Diagonals bisect each other at right angles.

(C) Diagonals bisect the angles at the vertices.

(D) Opposite angles are equal.

Question 6. To construct a square with a given side length 's', which methods are valid?

(A) Construct a line segment of length 's', then construct $90^\circ$ angles at both endpoints, and mark points at a distance 's' along the perpendiculars.

(B) Construct a line segment of length 's'. Construct a $90^\circ$ angle at one endpoint and mark a point at distance 's' on the arm. From the other two points, draw arcs of radius 's' to find the fourth vertex.

(C) Construct a rectangle with length and width equal to 's'.

(D) Construct a rhombus with side length 's' and one angle $90^\circ$.

Question 7. When constructing quadrilaterals, the process is often simplified by dividing the quadrilateral into triangles. Which construction criteria for triangles are commonly used as steps in quadrilateral constructions?

(A) SSS criterion (e.g., using a diagonal and two sides).

(B) SAS criterion (e.g., given two adjacent sides and the included angle).

(C) ASA criterion (e.g., given one side and two adjacent angles).

(D) RHS criterion (e.g., when right angles are involved, like in rectangles or squares).

Question 8. To construct a parallelogram given two adjacent sides and a diagonal, say sides a, b and diagonal d, which methods are valid?

(A) Construct a triangle with sides a, b, and d using SSS. Then use the parallelogram property that opposite sides are equal to locate the fourth vertex using arcs.

(B) Construct a triangle with sides a, b, and d. The constructed triangle forms one half of the parallelogram divided by the diagonal.

(C) Construct a triangle using SAS if the angle between a and b is known.

(D) Use the property that diagonals bisect each other (if diagonal lengths are known).

Question 9. To construct a rhombus given the lengths of its two diagonals, $d_1$ and $d_2$, which steps are correct?

(A) Draw one diagonal, say length $d_1$.

(B) Construct the perpendicular bisector of this diagonal.

(C) On the perpendicular bisector, mark points on either side of the midpoint at a distance of $d_2/2$.

(D) Join the endpoints of $d_1$ to the points marked on the perpendicular bisector to form the rhombus.

Question 10. Which of the following quadrilaterals always have diagonals that bisect each other at right angles?

(A) Parallelogram

(B) Rectangle

(C) Rhombus

(D) Square

Question 1. To construct a tangent to a circle at a point P on the circle, given the center O, which steps are correct?

(A) Draw the radius OP.

(B) Construct a line perpendicular to the radius OP at the point P.

(C) Extend the radius OP beyond P and construct a $90^\circ$ angle at P.

(D) The constructed perpendicular line is the tangent to the circle at P.

Question 2. The construction of a tangent at a point on the circle relies on which fundamental property?

(A) A tangent is a line that touches the circle at exactly one point.

(B) The radius drawn to the point of contact is perpendicular to the tangent at that point.

(C) The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

(D) All radii of a circle are equal in length.

Question 3. To construct tangents to a circle from a point P outside the circle with center O, which steps are involved?

(A) Join O to P.

(B) Find the midpoint M of the segment OP (e.g., by constructing the perpendicular bisector of OP).

(C) With M as centre, draw a circle with radius OM (or MP).

(D) The points where the circle with centre M intersects the given circle are the points of tangency.

Question 4. In the construction of tangents from an external point P to a circle with center O, why are the points of intersection of the circle with diameter OP and the original circle the points of tangency (say Q and R)?

(A) The angle $\angle OQP$ is subtended by the diameter OP in the circle centered at M.

(B) The angle subtended by a diameter at any point on the circumference is $90^\circ$, so $\angle OQP = 90^\circ$.

(C) OQ is a radius of the original circle.

(D) Since OQ is a radius and $\angle OQP = 90^\circ$, PQ is perpendicular to the radius at its endpoint on the circle, thus PQ is a tangent.

Question 5. Which statements are true about tangents drawn to a circle from an external point P?

(A) Exactly two tangents can be drawn from an external point.

(B) The lengths of the two tangent segments from P to the points of contact are equal.

(C) The line segment joining the center of the circle to the external point P bisects the angle between the two tangents.

(D) The line segment joining the center to the external point is perpendicular to the chord joining the points of contact.

Question 6. To construct a pair of tangents to a circle from an external point such that the angle between the tangents is a specific value $\theta$, which steps are useful?

(A) Calculate the angle between the radii to the points of contact using the property that $\angle AOB = 180^\circ - \theta$ in quadrilateral OAPB.

(B) Construct this calculated angle ($180^\circ - \theta$) at the center of the circle, with arms intersecting the circle at points A and B.

(C) At points A and B, construct lines perpendicular to the radii OA and OB respectively.

(D) The intersection point of these perpendicular lines is the external point P, and PA and PB are the required tangents.

Question 7. Justification for constructing tangents from an external point P often involves showing congruence. If O is the center, and Q, R are points of tangency, consider $\triangle OQP$ and $\triangle ORP$. Which congruence criterion can be used to show PQ = PR and that OP bisects $\angle QPR$ and $\angle QOR$?

(A) SSS

(B) SAS (OQ=OR, OP=OP, but the included angle at O is not known initially).

(C) ASA

(D) RHS (OQ = OR (radii), OP = OP (common hypotenuse), $\angle OQP = \angle ORP = 90^\circ$ (angle in semicircle construction)).

Question 8. Which statements are true about tangents to a circle?

(A) A tangent is always perpendicular to the radius at the point of tangency.

(B) From a point inside a circle, no tangent can be drawn.

(C) From a point on the circle, exactly one tangent can be drawn.

(D) From a point outside the circle, two tangents can be drawn.

Question 9. If two tangents from an external point P to a circle with center O meet at $90^\circ$, what is true about the quadrilateral OAPB (A, B are points of contact)?

(A) It is a square.

(B) It is a rectangle.

(C) OA = OB (radii), $\angle OAP = \angle OBP = 90^\circ$, $\angle APB = 90^\circ$. This implies $\angle AOB = 90^\circ$.

(D) Since all angles are $90^\circ$ and OA=OB (radii), OAPB is a square.

Question 10. The construction of a tangent at a point on the circle directly uses which other fundamental construction method?

(A) Constructing the perpendicular to a line from a point on the line.

(B) Constructing a $90^\circ$ angle.

(C) Copying a line segment.

(D) Bisecting an angle.

Question 11. If you are asked to construct tangents from an external point P to a circle but the center O is not given, what are some valid first steps that could lead to finding the tangents?

(A) Find the center of the circle (e.g., by finding the intersection of perpendicular bisectors of two non-parallel chords).

(B) Draw a secant through P intersecting the circle at two points and use properties of tangents and secants.

(C) Draw a line through P and try to guess the point of tangency.

(D) Construct a random line through P and check if it is perpendicular to the radius.

Question 12. In the construction of tangents from an external point P, the circle with diameter OP is drawn. Why is this circle important?

(A) It passes through the center O and the external point P.

(B) Its intersections with the original circle (say Q and R) create right angles $\angle OQP$ and $\angle ORP$.

(C) These right angles confirm that OQ is perpendicular to PQ and OR is perpendicular to PR.

(D) Its radius is OP.

Question 1. What is the primary role of justification in geometric constructions?

(A) To provide a mathematical proof that the constructed figure possesses the required geometric properties.

(B) To explain the sequence of steps performed.

(C) To ensure the drawing is accurate.

(D) To demonstrate understanding of the underlying geometric principles and theorems.

Question 2. Formal justification in geometric constructions typically involves applying which of the following?

(A) Basic definitions (e.g., what is a perpendicular line, what is an angle bisector).

(B) Axioms or postulates (statements accepted as true without proof, like Euclid's postulates).

(C) Previously proven theorems (like congruence criteria, BPT, properties of parallel lines, circle theorems).

(D) Measurements taken from the constructed figure.

Question 3. The justification for constructing an angle bisector involves proving the congruence of two triangles formed within the construction. Which specific congruence criterion is most commonly used?

(A) SAS

(B) SSS

(C) ASA

(D) RHS

Question 4. The justification for the perpendicular bisector construction often proves that any point on the constructed line is equidistant from the endpoints of the segment. This is often achieved by showing that triangles formed by connecting a point on the line to the segment endpoints are congruent. Which criterion is typically used?

(A) SSS

(B) SAS

(C) ASA

(D) AAS

Question 5. The justification for constructing parallel lines using corresponding or alternate interior angles relies on:

(A) The theorems stating the properties of angles formed by parallel lines and a transversal.

(B) The converse theorems: if corresponding/alternate interior angles are equal, the lines are parallel.

(C) The accuracy of copying the angles.

(D) The fact that parallel lines are equidistant.

Question 6. The justification for dividing a line segment in a given ratio m:n using the auxiliary ray and parallel lines primarily uses:

(A) Properties of similar triangles.

(B) The Basic Proportionality Theorem (BPT) or Thales Theorem.

(C) The property that parallel lines cut transversals proportionally.

(D) Congruence of the equal segments on the auxiliary ray.

Question 7. When justifying the construction of basic triangles (SSS, SAS, ASA), the primary tool used is:

(A) Similarity of triangles.

(B) The congruence criteria of triangles.

(C) Properties of quadrilaterals.

(D) The Pythagorean theorem.

Question 8. The justification for constructing tangents to a circle from an external point P involves proving that the constructed lines from P to the points of contact are indeed perpendicular to the radii at those points. This proof commonly utilizes:

(A) The property that the angle in a semicircle is a right angle.

(B) The congruence of triangles formed by the center, the external point, and the points of contact (e.g., $\triangle OQP \cong \triangle ORP$ by RHS).

(C) The definition of a tangent.

(D) The property that tangent lengths from an external point are equal.

Question 9. Verifying the accuracy of a geometric construction by measuring lengths and angles with tools like a ruler or protractor:

(A) Can help identify mistakes made during the physical drawing process.

(B) Provides a practical check on the outcome of the construction.

(C) Constitutes a formal mathematical proof or justification.

(D) Provides empirical evidence but is not considered a rigorous geometric justification based on axioms and theorems.

Question 10. Which of the following are considered fundamental building blocks of Euclidean geometry upon which proofs and justifications are built?

(A) Undefined terms (like point, line, plane).

(B) Definitions (explaining terms like angle, triangle, circle).

(C) Axioms or Postulates (assumed true statements).

(D) Theorems (statements proven from the fundamental assumptions).

Question 11. Justification helps to understand "why" a construction works, reinforcing the logical structure of geometry. It ensures that the construction is valid in theory, not just appears correct in practice.

(A) True

(B) False

(C) It connects the drawing steps to the underlying mathematical properties.

(D) It is primarily for academic exercises and not practical drawing.

Question 12. Formal justification of a construction ensures that the method will always produce the desired geometric figure with the required properties, provided the initial conditions are met and the steps are followed logically based on valid geometric principles.

(A) True

(B) False

(C) Justification only confirms accuracy for the specific example drawn.

(D) Justification relies on precise measurement.