Case Study / Scenario-Based MCQs for Sub-Topics of Topic 4: Geometry

Question 1. Imagine a map of Delhi. The location of India Gate is marked by a small dot. This dot represents which basic geometric element?

(A) A line

(B) A plane

(C) A point

(D) A line segment

Question 2. A straight road stretches from one city to another, continuing infinitely in both directions. This concept is best modeled by:

(A) A ray

(B) A line

(C) A line segment

(D) A curve

Question 3. Look at the surface of a smooth table top. It is a flat surface that extends without end. This surface is a representation of a:

(A) Line

(B) Point

(C) Plane

(D) Ray

Question 4. A piece of string is stretched tightly between two nails on a wall. The string between the two nails represents a:

(A) Ray

(B) Line

(C) Point

(D) Line segment

Question 5. Imagine a beam of light coming out of a torch. It starts at the torch and goes on and on in one direction. This is best modeled by a:

(A) Line segment

(B) Line

(C) Ray

(D) Curve

Question 6. Two railway tracks running side-by-side are designed so they never meet. In geometric terms, these tracks can be considered as:

(A) Intersecting lines

(B) Perpendicular lines

(C) Parallel lines

(D) Skew lines

Question 7. The hands of a clock meeting at the center when forming angles less than 180 degrees can be thought of as representing:

(A) Two rays from a common point

(B) Two line segments from a common point

(C) Two lines from a common point

(D) A point and two rays

Question 1. Ramesh wants to measure the length of his desk. He uses a 30 cm ruler. He places the ruler along the edge and finds the desk is longer than the ruler. What should he do to find the total length?

(A) Measure the angle of the desk corners.

(B) Estimate the remaining length.

(C) Mark the end of the ruler and move it along, adding the measurements.

(D) Use a protractor instead.

Question 2. A carpenter needs to cut a wooden board at a precise angle for a furniture joint. Which tool is essential for him to measure this angle accurately?

(A) Measuring tape

(B) Ruler

(C) Protractor

(D) Compass

Question 3. Priya and Anjali are comparing the lengths of two pencils. Priya says her pencil is longer. To verify this without breaking or marking the pencils, they can use a ruler. What are they comparing when they use the ruler?

(A) The weight of the pencils.

(B) The colour of the pencils.

(C) The numerical value of the distance between the two ends of each pencil.

(D) The angles formed by the pencils.

Question 4. The hands of a clock at 10:00 form an angle. What are the parts of this angle in geometric terms?

(A) The numbers 10 and 12 are the arms, and the centre is the vertex.

(B) The hour hand and the minute hand are the arms, and the centre is the vertex.

(C) The hour hand is one arm, the number 10 is the vertex, and the minute hand is the other arm.

(D) The angle itself is the vertex, and the hands are the measure.

Question 5. A student is asked to measure the angle formed by two edges of a book cover. They place the protractor correctly. The angle is measured to be $90^\circ$. This is a standard measure for:

(A) A straight angle.

(B) An acute angle.

(C) A right angle.

(D) An obtuse angle.

Question 6. A point is located inside an angle, not on the arms or the vertex. Where is this point located?

(A) On the vertex of the angle.

(B) In the exterior of the angle.

(C) On an arm of the angle.

(D) In the interior of the angle.

Question 1. Look at the roof of a typical house. The angle where two sloping parts of the roof meet at the top is often less than $90^\circ$. This type of angle is a(n):

(A) Right angle

(B) Obtuse angle

(C) Acute angle

(D) Straight angle

Question 2. A door opens outwards. When it is opened just a little, the angle formed by the door and the wall is small. As it opens wider, the angle increases. If the door is fully opened to lie flat against the wall, the angle is $180^\circ$. This angle is a(n):

(A) Right angle

(B) Complete angle

(C) Straight angle

(D) Reflex angle

Question 3. A traffic intersection where two roads cross each other at perfect right angles is an example of:

(A) Parallel lines

(B) Skew lines

(C) Perpendicular lines

(D) Curved lines

Question 4. A geometry student is asked to find the set of all points that are equidistant from the two endpoints of a line segment AB. The teacher hints that the locus of these points forms a line that is perpendicular to AB and passes through its middle. This line is the:

(A) Median

(B) Altitude

(C) Angle bisector

(D) Perpendicular bisector

Question 5. Consider an angle that measures $105^\circ$. This angle is classified as:

(A) Acute angle

(B) Right angle

(C) Obtuse angle

(D) Reflex angle

Question 6. When you make a full turn ($360^\circ$) while facing a direction, you return to your starting orientation. This represents a:

(A) Zero angle

(B) Straight angle

(C) Reflex angle

(D) Complete angle

Question 1. Two angles add up to exactly $90^\circ$. These angles are called:

(A) Supplementary angles

(B) Adjacent angles

(C) Complementary angles

(D) Linear pair

Question 2. Two angles $\angle A$ and $\angle B$ are adjacent, sharing a common vertex and arm. Their non-common arms extend in opposite directions, forming a straight line. What type of pair is this?

(A) Complementary angles

(B) Supplementary angles

(C) Vertically opposite angles

(D) A linear pair

Question 3. Two lines intersect in the middle of a page. The angles directly across from each other at the intersection point are always equal. These angles are called:

(A) Adjacent angles

(B) Corresponding angles

(C) Linear pair angles

(D) Vertically opposite angles

Question 4. If an angle measures $40^\circ$, what is the measure of its supplementary angle?

(A) $50^\circ$

(B) $140^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 5. Consider two adjacent angles $\angle PQR$ and $\angle RQS$ that form a right angle $\angle PQS$. What is the relationship between $\angle PQR$ and $\angle RQS$?

(A) They form a linear pair.

(B) They are supplementary.

(C) They are vertically opposite.

(D) They are complementary.

Question 6. If two lines intersect and one angle formed is $85^\circ$, what is the measure of the angle vertically opposite to it?

(A) $95^\circ$

(B) $85^\circ$

(C) $10^\circ$

(D) $180^\circ$

Question 1. Imagine two straight roads running parallel to each other, intersected by a diagonal road. The diagonal road is a:

(A) Parallel line

(B) Perpendicular line

(C) Transversal

(D) Ray

Question 2. When a transversal intersects two lines, it creates eight angles. The angles that are on the same side of the transversal and in corresponding positions (one inside, one outside) are called:

(A) Alternate interior angles

(B) Consecutive interior angles

(C) Corresponding angles

(D) Vertically opposite angles

Question 3. If a transversal intersects two parallel lines, and one of the interior angles on the same side of the transversal is $65^\circ$, what is the measure of the other interior angle on the same side?

(A) $65^\circ$

(B) $115^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 4. A builder wants to check if two walls are parallel. He measures the angle formed by a crossing beam with each wall on the interior and on the same side. If these two angles add up to $180^\circ$, what can he conclude about the walls?

(A) They are perpendicular.

(B) They are intersecting.

(C) They are parallel.

(D) They are skew.

Question 5. Look at a window frame with horizontal and vertical bars. The horizontal bars are parallel to each other. If a vertical bar crosses them, it acts as a transversal. What is the relationship between the angles formed by the vertical bar and the upper horizontal bar on the interior and opposite sides?

(A) They are supplementary.

(B) They are complementary.

(C) They are equal (alternate interior angles).

(D) They form a linear pair.

Question 6. Lines $l$ and $m$ are intersected by a transversal $t$. If a pair of corresponding angles are not equal, what does this tell us about lines $l$ and $m$?

(A) They are parallel.

(B) They are perpendicular.

(C) They will intersect somewhere.

(D) They are coincident.

Question 1. When constructing a geometric proof, a student states that "Through the two given points, a unique straight line can be drawn." This statement is an example of:

(A) A definition

(B) An undefined term

(C) A postulate

(D) A theorem that needs to be proven

Question 2. A mathematician is working on proving a new geometric property. They start with statements that are accepted as true without proof, like "Things which are equal to the same thing are equal to one another." These statements are called:

(A) Definitions

(B) Undefined terms

(C) Axioms (Common Notions)

(D) Theorems

Question 3. Consider Euclid's statement: "All right angles are equal to one another." This ensures consistency in angle measurement and is classified as a:

(A) Definition

(B) Postulate

(C) Axiom

(D) Theorem

Question 4. A geometry student is asked to prove that the sum of angles in a triangle is $180^\circ$. They must use definitions, axioms, and postulates to build a logical argument. The statement they are proving is a:

(A) Definition

(B) Postulate

(C) Undefined term

(D) Theorem

Question 5. Historically, Euclid's Fifth Postulate was controversial. Mathematicians tried for centuries to prove it from the other postulates but failed. Eventually, this led to the development of geometries where this postulate is false. What is this postulate about?

(A) Drawing a straight line between two points.

(B) Extending a line segment indefinitely.

(C) The existence and uniqueness of parallel lines.

(D) The equality of right angles.

Question 6. In a proof, a step uses the justification "Things which coincide with one another are equal to one another". This is a basic assumption from Euclid's Common Notions. What does it imply?

(A) Any two points can define a line.

(B) If two figures can be perfectly superimposed, they are equal in all respects.

(C) Parallel lines never meet.

(D) A part is always smaller than the whole.

Question 1. A child draws a simple closed shape using three straight sticks. The shape they form is a type of polygon. What is the minimum number of sides a polygon can have?

(A) 2

(B) 3

(C) 4

(D) 5

Question 2. Look at the shape of a standard ₹ 5 coin. The boundary is a simple closed curve. Is the boundary of a ₹ 5 coin a polygon?

(A) Yes, it is a polygon with many sides.

(B) No, because it is a curved boundary, not made of line segments.

(C) Yes, because it is a closed shape.

(D) Yes, because it is simple.

Question 3. A student draws a pentagon where one of the interior angles is greater than $180^\circ$. This type of polygon is called a:

(A) Convex polygon

(B) Regular polygon

(C) Concave polygon

(D) Equilateral polygon

Question 4. A geometric shape has all its sides equal and all its interior angles equal. This describes a:

(A) Convex polygon

(B) Concave polygon

(C) Irregular polygon

(D) Regular polygon

Question 5. A quadrilateral is a polygon. What is the sum of the interior angles of any convex quadrilateral?

(A) $180^\circ$

(B) $360^\circ$

(C) $540^\circ$

(D) Depends on the side lengths

Question 6. How many diagonals can be drawn from one vertex of a hexagon?

(A) 2

(B) 3

(C) 4

(D) 5

Question 1. A triangle has side lengths 4 cm, 6 cm, and 8 cm. How would you classify this triangle based on its sides?

(A) Isosceles triangle

(B) Equilateral triangle

(C) Scalene triangle

(D) Right-angled triangle

Question 2. A triangle has angles measuring $30^\circ$, $60^\circ$, and $90^\circ$. How would you classify this triangle based on its angles?

(A) Acute-angled triangle

(B) Obtuse-angled triangle

(C) Right-angled triangle

(D) Equiangular triangle

Question 3. A triangle has two sides of equal length. This triangle is called a(n):

(A) Scalene triangle

(B) Equilateral triangle

(C) Isosceles triangle

(D) Right-angled triangle

Question 4. Can a triangle have angles $100^\circ, 40^\circ, 40^\circ$? If yes, what type of triangle is it?

(A) Yes, it's an acute-angled triangle.

(B) Yes, it's an obtuse-angled triangle.

(C) Yes, it's a right-angled triangle.

(D) No, these angles cannot form a triangle.

Question 5. What is the classification of a triangle based on angles if all three angles are less than $90^\circ$?

(A) Right-angled

(B) Obtuse-angled

(C) Acute-angled

(D) Straight-angled

Question 6. A triangle has side lengths 7 cm, 7 cm, and 7 cm. How is this triangle classified based on its sides?

(A) Scalene triangle

(B) Isosceles triangle

(C) Equilateral triangle

(D) Both B and C

Question 1. In $\triangle$ABC, $\angle A = 50^\circ$ and $\angle B = 60^\circ$. What is the measure of $\angle C$?

(A) $70^\circ$

(B) $110^\circ$

(C) $180^\circ$

(D) $130^\circ$

Question 2. In $\triangle$PQR, the exterior angle at Q is $110^\circ$. If $\angle P = 70^\circ$, what is the measure of $\angle R$?

(A) $40^\circ$

(B) $50^\circ$

(C) $60^\circ$

(D) $110^\circ$

Question 3. An architect is designing a triangular support structure. The sides are 5 metres, 10 metres, and 12 metres. Can these lengths form a triangle?

(A) Yes, because $5+10 > 12$.

(B) No, because $5+10$ is not greater than $12$.

(C) Yes, because all lengths are positive.

(D) Yes, but only if it's a right triangle.

Question 4. In $\triangle$XYZ, side XY = 10 cm, YZ = 15 cm, and XZ = 8 cm. Based on side-angle relationships, which is the largest angle?

(A) $\angle X$ (opposite YZ)

(B) $\angle Y$ (opposite XZ)

(C) $\angle Z$ (opposite XY)

(D) Cannot be determined without angle measures

Question 5. A triangular park has angles $55^\circ$, $55^\circ$, and $70^\circ$. What can you conclude about the sides of this park?

(A) All three sides are equal.

(B) Exactly two sides are equal.

(C) All three sides are different lengths.

(D) The park is a right-angled triangle.

Question 6. In $\triangle$LMN, $\angle L = 80^\circ$, $\angle M = 40^\circ$. Which side is the shortest?

(A) LM (opposite $\angle N = 180 - 80 - 40 = 60^\circ$)

(B) MN (opposite $\angle L = 80^\circ$)

(C) LN (opposite $\angle M = 40^\circ$)

(D) Cannot be determined

Question 1. A ladder 10 metres long is leaning against a vertical wall. The base of the ladder is 6 metres away from the wall on the ground. How high up the wall does the ladder reach?

(A) 4 metres

(B) 8 metres

(C) 16 metres

(D) $\sqrt{136}$ metres

Question 2. A triangle has side lengths 7 cm, 24 cm, and 25 cm. Is this a right-angled triangle?

(A) Yes, because $7+24 > 25$.

(B) No, it's an acute-angled triangle.

(C) Yes, because $7^2 + 24^2 = 25^2$.

(D) Yes, because it forms a triangle.

Question 3. A carpenter is building a rectangular door frame. The sides measure 0.9 metres and 2 metres. To ensure the frame is perfectly rectangular (has $90^\circ$ corners), he measures the diagonal. What should be the length of the diagonal?

(A) $2.9$ metres

(B) $\sqrt{0.9^2 + 2^2}$ metres

(C) $\sqrt{0.81 + 4}$ metres

(D) Both B and C

Question 4. A triangle has side lengths 6 cm, 8 cm, and 11 cm. Is this a right-angled triangle?

(A) Yes, because $6^2 + 8^2 = 11^2$.

(B) No, because $36 + 64 = 100 \neq 121$.

(C) Yes, because it is an integer triplet.

(D) Cannot be determined.

Question 5. A right-angled triangle has a hypotenuse of length 13 units and one leg of length 5 units. What is the length of the other leg?

(A) 8 units

(B) 12 units

(C) $\sqrt{169+25}$ units

(D) $\sqrt{13^2 - 5^2}$ units

Question 6. A triangle has side lengths 2, 3, and 4. Is it a right triangle? If not, what type is it based on the angle opposite the longest side?

(A) Right triangle.

(B) Acute-angled triangle ($2^2 + 3^2 = 4+9=13 > 4^2=16$ is false).

(C) Obtuse-angled triangle ($2^2 + 3^2 = 13 < 4^2=16$).

(D) Cannot form a triangle.

Question 1. A stationery shop sells two identical protractors. If you place one exactly on top of the other, they match perfectly. These two protractors are:

(A) Similar only

(B) Congruent

(C) Different shapes

(D) Different sizes

Question 2. Two triangles $\triangle \text{ABC}$ and $\triangle \text{XYZ}$ are given. If AB = XY, BC = YZ, and CA = ZX, by which congruence criterion can we say $\triangle \text{ABC} \cong \triangle \text{XYZ}$?

(A) ASA

(B) SAS

(C) SSS

(D) AAS

Question 3. In $\triangle \text{PQR}$ and $\triangle \text{STU}$, if PQ = ST, $\angle Q = \angle T$, and QR = TU, by which congruence criterion can we say $\triangle \text{PQR} \cong \triangle \text{STU}$?

(A) ASA

(B) SAS

(C) AAS

(D) RHS

Question 4. If $\triangle \text{LMN} \cong \triangle \text{PQR}$, and you want to conclude that $\angle M = \angle Q$, which principle are you using?

(A) Angle Sum Property

(B) Side-Angle Relationship

(C) CPCTC

(D) Triangle Inequality

Question 5. Two right-angled triangles are given. $\triangle \text{ABC}$ is right-angled at B, and $\triangle \text{DEF}$ is right-angled at E. If hypotenuse AC = hypotenuse DF and leg AB = leg DE, by which criterion are the triangles congruent?

(A) SAS

(B) AAS

(C) ASA

(D) RHS

Question 6. Are two squares with different side lengths congruent?

(A) Yes, they have the same shape.

(B) No, because they have different sizes.

(C) Yes, they are similar and hence congruent.

(D) Cannot be determined.

Question 1. A photograph is printed in two different sizes, a small one and a large one. The shapes are the same, but the sizes are different. These two photographs are:

(A) Congruent

(B) Similar

(C) Neither congruent nor similar

(D) Identical

Question 2. $\triangle \text{ABC}$ has angles $40^\circ, 60^\circ, 80^\circ$. $\triangle \text{PQR}$ has angles $40^\circ, 60^\circ, 80^\circ$. Are these triangles similar?

(A) Yes, by AA similarity (or AAA).

(B) No, their side lengths might be different.

(C) Yes, by SSS similarity.

(D) Only if their areas are equal.

Question 3. In $\triangle \text{ABC}$, a line DE is drawn parallel to BC, intersecting AB at D and AC at E. According to the Basic Proportionality Theorem, which ratio is equal to AE/EC?

(A) AD/AB

(B) DB/AD

(C) AD/DB

(D) DE/BC

Question 4. Are two circles of different radii similar?

(A) Yes, because they have the same shape.

(B) No, because their sizes are different.

(C) Yes, they are also congruent.

(D) Only if their centres are the same.

Question 5. A student draws a line segment connecting two points D and E on sides AB and AC of $\triangle \text{ABC}$. They measure AD/DB and AE/EC and find that the ratios are equal. What can they conclude about the line segment DE?

(A) DE is perpendicular to BC.

(B) DE is congruent to BC.

(C) DE is parallel to BC.

(D) D and E are midpoints.

Question 6. $\triangle \text{XYZ}$ has sides 4 cm, 6 cm, 8 cm. $\triangle \text{PQR}$ has sides 2 cm, 3 cm, 4 cm. Are these triangles similar?

(A) Yes, by AA similarity.

(B) Yes, by SSS similarity (check proportionality).

(C) No, the angles are different.

(D) Cannot be determined.

Question 1. Two similar triangles have corresponding sides in the ratio 1:2. What is the ratio of their areas?

(A) 1:2

(B) 1:4

(C) 2:1

(D) 4:1

Question 2. The areas of two similar triangles are $16 \text{ cm}^2$ and $25 \text{ cm}^2$. What is the ratio of their corresponding sides?

(A) 16:25

(B) 4:5

(C) $\sqrt{16}:\sqrt{25}$

(D) Both B and C

Question 3. A map of a city is drawn to a scale of 1 cm : 100 metres. A triangular park on the map has an area of $3 \text{ cm}^2$. What is the actual area of the park?

(A) $300 \text{ m}^2$

(B) $3000 \text{ m}^2$

(C) $30000 \text{ m}^2$

(D) $300000 \text{ m}^2$

Question 4. In a right-angled triangle ABC, right-angled at B, an altitude BD is drawn to the hypotenuse AC. $\triangle \text{ADB}$ is similar to $\triangle \text{ABC}$. If AD = 4 cm and AB = 6 cm, what is the length of AC?

(A) 9 cm

(B) 13 cm

(C) 10 cm

(D) 8 cm

Question 5. A flagpole casts a shadow 15 metres long. At the same time, a 2-metre tall person casts a shadow 3 metres long. Assuming the sun's rays are parallel, what is the height of the flagpole?

(A) 10 metres

(B) 22.5 metres

(C) $\frac{15 \times 2}{3}$ metres

(D) Both A and C

Question 6. If two similar triangles have perimeters in the ratio 5:6, what is the ratio of their areas?

(A) 5:6

(B) 25:36

(C) $\sqrt{5}:\sqrt{6}$

(D) 6:5

Question 1. A gate is built in the shape of a quadrilateral. It has two pairs of adjacent sides equal, but its opposite sides are not equal. What type of quadrilateral is this?

(A) Parallelogram

(B) Rhombus

(C) Kite

(D) Trapezium

Question 2. A field is in the shape of a parallelogram. If one of its angles is $75^\circ$, what are the measures of the other three angles?

(A) $75^\circ, 105^\circ, 105^\circ$

(B) $105^\circ, 75^\circ, 105^\circ$

(C) $75^\circ, 90^\circ, 110^\circ$

(D) $105^\circ, 105^\circ, 75^\circ$

Question 3. A board is in the shape of a rectangle. The diagonals are drawn. What property do the diagonals of a rectangle have?

(A) They are perpendicular.

(B) They bisect the angles.

(C) They are equal and bisect each other.

(D) They are perpendicular bisectors of each other (only for a square).

Question 4. A student draws a quadrilateral ABCD. They find that the diagonals AC and BD bisect each other. What type of quadrilateral is ABCD?

(A) Trapezium

(B) Kite

(C) Parallelogram

(D) Rhombus (only if diagonals are perpendicular)

Question 5. A traffic sign is shaped like a rhombus. One of its angles is $60^\circ$. What are the measures of the other angles?

(A) $60^\circ, 120^\circ, 120^\circ$

(B) $60^\circ, 60^\circ, 120^\circ$

(C) $120^\circ, 60^\circ, 120^\circ$

(D) $90^\circ, 90^\circ, 90^\circ$

Question 6. Which of the following statements is true about a square?

(A) It is a rectangle but not a rhombus.

(B) It is a rhombus but not a rectangle.

(C) It is both a rectangle and a rhombus.

(D) It is a parallelogram but neither a rectangle nor a rhombus.

Question 1. In $\triangle$ABC, D is the midpoint of AB and E is the midpoint of AC. If the length of BC is 14 cm, what is the length of DE?

(A) 7 cm

(B) 14 cm

(C) 28 cm

(D) Depends on the angles of the triangle

Question 2. In $\triangle$PQR, M is the midpoint of PQ. A line is drawn through M parallel to QR, intersecting PR at N. What can you conclude about point N?

(A) N is the midpoint of PQ.

(B) N is the midpoint of QR.

(C) N is the midpoint of PR.

(D) MN = $\frac{1}{2}$ QR (This is a consequence, not the conclusion about N).

Question 3. A quadrilateral ABCD is given. P, Q, R, and S are the midpoints of the sides AB, BC, CD, and DA respectively. What type of quadrilateral is PQRS?

(A) Square

(B) Rectangle

(C) Rhombus

(D) Parallelogram

Question 4. In $\triangle$XYZ, the segment joining the midpoints of XY and XZ is parallel to YZ. This statement is based on:

(A) The converse of the Mid-Point Theorem.

(B) The Mid-Point Theorem itself.

(C) Basic Proportionality Theorem.

(D) Similarity of triangles.

Question 5. In $\triangle$LMN, P, Q, R are the midpoints of LM, MN, NL respectively. If the perimeter of $\triangle$LMN is 30 cm, what is the perimeter of $\triangle$PQR?

(A) 10 cm

(B) 15 cm

(C) 20 cm

(D) 30 cm

Question 6. If the figure formed by joining the midpoints of a quadrilateral's sides is a rectangle, what can be concluded about the diagonals of the original quadrilateral?

(A) They are equal.

(B) They are perpendicular.

(C) They are parallel.

(D) They bisect each other.

Question 1. A farmer has a rectangular field and a parallelogram-shaped field. Both fields are on the same base and between the same two parallel fence lines. Which field has a larger area?

(A) The rectangular field

(B) The parallelogram field

(C) They have equal area

(D) Cannot be determined without dimensions

Question 2. A carpenter cuts a triangular piece of wood. He then cuts another triangular piece from the same board, such that it has the same base length and the same height as the first triangle. What is true about the areas of these two triangular pieces?

(A) The areas are different.

(B) The areas are equal.

(C) One area is half the other.

(D) The triangles must be congruent.

Question 3. In parallelogram ABCD, the diagonal AC is drawn. This divides the parallelogram into two triangles, $\triangle \text{ABC}$ and $\triangle \text{ADC}$. What is the relationship between their areas?

(A) Area($\triangle \text{ABC}$) > Area($\triangle \text{ADC}$)

(B) Area($\triangle \text{ABC}$) < Area($\triangle \text{ADC}$)

(C) Area($\triangle \text{ABC}$) = Area($\triangle \text{ADC}$)

(D) Cannot be determined

Question 4. In $\triangle$PQR, S is the midpoint of QR. The line segment PS is a median. What is the relationship between the areas of $\triangle \text{PQS}$ and $\triangle \text{PRS}$?

(A) Area($\triangle \text{PQS}$) > Area($\triangle \text{PRS}$)

(B) Area($\triangle \text{PQS}$) < Area($\triangle \text{PRS}$)

(C) Area($\triangle \text{PQS}$) = Area($\triangle \text{PRS}$)

(D) $\triangle \text{PQS}$ is congruent to $\triangle \text{PRS}$

Question 5. Two triangles are drawn on the same base. If they have equal areas, what can you say about their third vertices (the vertices opposite the common base)?

(A) They must be the same point.

(B) They must lie on a line perpendicular to the base.

(C) They must lie on a line parallel to the base.

(D) They must be on the base line itself.

Question 6. If the area of a parallelogram is $40 \text{ cm}^2$, and a triangle shares the same base and lies between the same parallels as the parallelogram, what is the area of the triangle?

(A) $40 \text{ cm}^2$

(B) $80 \text{ cm}^2$

(C) $20 \text{ cm}^2$

(D) Cannot be determined

Question 1. You are drawing a circle using a compass. The point where you place the compass needle is the:

(A) Radius

(B) Diameter

(C) Circumference

(D) Centre

Question 2. The distance from the centre of a circular rangoli design to its edge is 1 metre. This distance is the rangoli's:

(A) Diameter

(B) Chord

(C) Radius

(D) Circumference

Question 3. A straight line segment connects two points on the boundary of a circular pond, but it does not pass through the exact centre. This line segment is a:

(A) Radius

(B) Diameter

(C) Chord

(D) Tangent

Question 4. The boundary of a circular swimming pool has a length of 50 metres. This length is the pool's:

(A) Area

(B) Diameter

(C) Radius

(D) Circumference

Question 5. A slice of pizza cut from the centre to the crust edge is in the shape of a sector of a circle. What geometric parts form the boundary of this sector?

(A) Two chords and an arc

(B) Two radii and an arc

(C) Two radii and a chord

(D) One radius and two arcs

Question 6. You have two ₹ 1 coins and one ₹ 10 coin. What can you say about the relationship between the two ₹ 1 coins compared to the ₹ 1 coin and the ₹ 10 coin in terms of basic circle properties?

(A) The two ₹ 1 coins are similar, and the ₹ 1 and ₹ 10 coins are congruent.

(B) The two ₹ 1 coins are congruent, and the ₹ 1 and ₹ 10 coins are similar.

(C) All three coins are congruent.

(D) All three coins are similar but not congruent.

Question 1. A chord AB in a circle subtends an angle of $80^\circ$ at the centre. What angle does it subtend at any point C on the major arc AB?

(A) $80^\circ$

(B) $160^\circ$

(C) $40^\circ$

(D) $100^\circ$

Question 2. A triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle. What type of triangle is this always?

(A) Acute-angled triangle

(B) Obtuse-angled triangle

(C) Right-angled triangle

(D) Equilateral triangle

Question 3. In a circle, two chords are of equal length. What can you say about their distances from the centre?

(A) They are at different distances.

(B) They are equidistant from the centre.

(C) They are parallel.

(D) They are perpendicular.

Question 4. Chord PQ subtends an angle of $50^\circ$ at a point R on the major arc. What angle does PQ subtend at another point S on the same major arc?

(A) $50^\circ$

(B) $100^\circ$

(C) $25^\circ$

(D) $130^\circ$

Question 5. A line segment from the centre of a circle is drawn perpendicular to a chord AB. If the length of the chord is 10 cm, what is the length of each part of the chord after being divided by this perpendicular?

(A) 10 cm

(B) 5 cm

(C) 20 cm

(D) Cannot be determined

Question 6. An arc subtends a reflex angle of $240^\circ$ at the centre. What angle does the same arc subtend at any point on the remaining part of the circle?

(A) $120^\circ$

(B) $60^\circ$

(C) $240^\circ$

(D) $30^\circ$

Question 1. A quadrilateral ABCD is drawn such that all its four vertices A, B, C, and D lie on a circle. What is this quadrilateral called?

(A) Parallelogram

(B) Rhombus

(C) Cyclic quadrilateral

(D) Trapezium

Question 2. In a cyclic quadrilateral PQRS, if $\angle P = 85^\circ$, what is the measure of $\angle R$?

(A) $85^\circ$

(B) $95^\circ$

(C) $180^\circ$

(D) $275^\circ$

Question 3. A student draws a quadrilateral ABCD and finds that $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$. What can the student conclude about this quadrilateral?

(A) It is a parallelogram.

(B) It is a rectangle.

(C) It is a cyclic quadrilateral.

(D) It is a rhombus.

Question 4. If a cyclic quadrilateral ABCD has $\angle A = 70^\circ$ and $\angle B = 100^\circ$, what are the measures of $\angle C$ and $\angle D$ respectively?

(A) $\angle C = 110^\circ, \angle D = 80^\circ$

(B) $\angle C = 80^\circ, \angle D = 110^\circ$

(C) $\angle C = 70^\circ, \angle D = 100^\circ$

(D) $\angle C = 100^\circ, \angle D = 70^\circ$

Question 5. A cyclic quadrilateral ABCD is given. If side AB is extended to a point E, what is the relationship between the exterior angle $\angle CBE$ and the interior angle $\angle D$?

(A) $\angle CBE + \angle D = 180^\circ$

(B) $\angle CBE = \angle D$

(C) $\angle CBE = \angle A$

(D) $\angle CBE + \angle C = 180^\circ$

Question 6. Can a parallelogram, which is not a rectangle, be a cyclic quadrilateral?

(A) Yes, always.

(B) No, because its angles are not all supplementary.

(C) Yes, if its diagonals are perpendicular.

(D) Only if its adjacent angles are equal.

Question 1. A straight line is drawn such that it touches a circular park boundary at exactly one point. This line is called a:

(A) Secant

(B) Chord

(C) Diameter

(D) Tangent

Question 2. A line passes through the interior of a circle and intersects the circle at two distinct points. This line is a:

(A) Tangent

(B) Chord

(C) Secant

(D) Radius

Question 3. From a point P located outside a circular wall, two lines are drawn that just touch the wall at points A and B. What is the relationship between the lengths of the segments PA and PB?

(A) PA > PB

(B) PA < PB

(C) PA = PB

(D) Cannot be determined

Question 4. A radius is drawn from the centre O to the point of contact A of a tangent line. What is the angle formed between the radius OA and the tangent line at A?

(A) $0^\circ$

(B) $45^\circ$

(C) $90^\circ$

(D) $180^\circ$

Question 5. How many tangent lines can be drawn to a circle from a point located on the circle's circumference?

(A) Zero

(B) One

(C) Two

(D) Infinite

Question 6. Two circles touch each other externally at a point. The distance between their centres is 10 cm. If the radius of one circle is 4 cm, what is the radius of the other circle?

(A) 6 cm

(B) 14 cm

(C) 2 cm

(D) 10 cm

Question 1. You hold a mirror along the vertical midline of the letter 'A'. The reflection in the mirror looks exactly like the other half of the 'A'. This shows that the letter 'A' has:

(A) Rotational symmetry

(B) Point symmetry

(C) Line symmetry

(D) Translational symmetry

Question 2. A square is folded in half along its diagonal. The two halves perfectly match. The diagonal is acting as the:

(A) Centre of rotation

(B) Angle of symmetry

(C) Axis of symmetry

(D) Line segment

Question 3. The letter 'S' when reflected across a vertical line does not look like an 'S'. However, the letter 'O' looks exactly like an 'O' when reflected across a vertical line. Which type of symmetry does 'O' have that 'S' lacks?

(A) Rotational symmetry

(B) Horizontal line symmetry

(C) Vertical line symmetry

(D) Point symmetry

Question 4. A point P has coordinates (3, 5). It is reflected across the x-axis to get point P'. What are the coordinates of P'?

(A) (-3, 5)

(B) (3, -5)

(C) (-3, -5)

(D) (5, 3)

Question 5. How many lines of symmetry does a rectangle that is NOT a square have?

(A) 0

(B) 1

(C) 2

(D) 4

Question 6. A designer creates a logo that is perfectly symmetrical when a mirror is placed along a certain line through it. This type of symmetry is called:

(A) Rotational symmetry

(B) Translational symmetry

(C) Reflectional symmetry

(D) Glide reflection

Question 1. A fan has 3 identical blades. When the fan is rotated by a certain angle, the blades appear in the same position as before the rotation. What is the smallest angle of rotation (other than $0^\circ$) by which this fan will coincide with itself?

(A) $180^\circ$

(B) $120^\circ$

(C) $90^\circ$

(D) $60^\circ$

Question 2. The symbol for the Swastika ($\Huge{\swarrow}$) has rotational symmetry. If rotated about its centre, it looks the same multiple times in a full turn. What is the order of rotational symmetry of the Swastika?

(A) 1

(B) 2

(C) 3

(D) 4

Question 3. A rectangle (that is not a square) is rotated about the intersection point of its diagonals. By what minimum angle (greater than $0^\circ$) will it coincide with its original position?

(A) $90^\circ$

(B) $180^\circ$

(C) $270^\circ$

(D) $360^\circ$

Question 4. A figure has rotational symmetry of order 5. What is the smallest angle of rotation?

(A) $360^\circ$

(B) $180^\circ$

(C) $90^\circ$

(D) $72^\circ$

Question 5. Which of the following letters of the English alphabet has rotational symmetry of order 2?

(A) A

(B) B

(C) C

(D) N

Question 6. If a shape only looks the same after a full $360^\circ$ rotation, what is its order of rotational symmetry?

(A) 0

(B) 1

(C) Infinite

(D) Undefined

Question 1. Which of the following is an example of a 3-dimensional shape?

(A) A triangle drawn on paper.

(B) The surface of a table.

(C) A ball.

(D) A shadow on the ground.

Question 2. A standard die used in games is a cube. How many faces does a cube have?

(A) 4

(B) 6

(C) 8

(D) 12

Question 3. The shape of a tin can used for packaging food is typically a cylinder. What kind of faces does a cylinder have?

(A) All plane faces.

(B) All curved faces.

(C) Plane and curved faces.

(D) No faces.

Question 4. A traffic cone used on roads has a circular base and tapers to a point. What is this shape called?

(A) Cylinder

(B) Sphere

(C) Pyramid

(D) Cone

Question 5. The corners of a room, where three walls meet, represent the vertices of a cuboid. How many vertices does a cuboid have?

(A) 4

(B) 6

(C) 8

(D) 12

Question 6. A standard Indian sweet box is often a cuboid. How many edges does a cuboid have?

(A) 6

(B) 8

(C) 10

(D) 12

Question 1. An engineer is drawing a sketch of a machine part. They use a drawing where the front face is drawn as it appears, but the lines going back into the page are drawn at an angle and sometimes shorter than their actual length to give a 3D effect. This is likely a(n):

(A) Isometric sketch

(B) Orthographic projection

(C) Oblique sketch

(D) Cross-section

Question 2. Imagine slicing a loaf of bread horizontally. The shape of the slice is the same as the shape of the end of the loaf. This shape is a:

(A) Net

(B) View

(C) Cross-section

(D) Surface

Question 3. If you slice a cucumber straight down the middle, you get a circular shape on the cut surface. This is a cross-section of the cucumber (assuming it's cylindrical). If you sliced it diagonally, what shape might you get?

(A) Circle

(B) Rectangle

(C) Ellipse

(D) Triangle

Question 4. An architect draws the floor plan of a building. This is essentially a view of the building from directly above. This type of drawing is called a:

(A) Front view

(B) Side view

(C) Top view

(D) Isometric view

Question 5. A graphic designer uses an isometric grid to draw a box. In this drawing, lines that are parallel in 3D space are drawn as parallel lines on the 2D paper. This is a characteristic of:

(A) Oblique sketches

(B) Orthographic projections

(C) Isometric sketches

(D) Perspective drawings

Question 6. You are given a net of a cube, which is a 2D pattern. What can you do with this net?

(A) Measure its area to find the cube's volume.

(B) Fold it along the lines to form the 3D cube.

(C) Cut it to find a cross-section of the cube.

(D) View it from different angles to visualize the cube.

Question 1. Which of the following solid shapes is NOT a polyhedron?

(A) A cube

(B) A triangular prism

(C) A square pyramid

(D) A sphere

Question 2. A student is given a list of numbers representing the vertices (V), edges (E), and faces (F) of a solid. They are told to check if it could be a convex polyhedron using Euler's formula. Which formula should they use?

(A) V + E + F = 2

(B) V - E + F = 2

(C) E - V + F = 2

(D) V + E - F = 2

Question 3. A prism has a pentagonal base. It has 7 faces and 15 edges. How many vertices does this pentagonal prism have?

(A) 8

(B) 9

(C) 10

(D) 12

Question 4. A polyhedron has 8 vertices and 12 faces. How many edges does it have, assuming it is convex and fits Euler's formula?

(A) 16

(B) 18

(C) 20

(D) 22

Question 5. A Platonic solid is a special type of convex polyhedron. One characteristic is that all its faces are the same congruent regular polygon. Which of the following is NOT a Platonic solid?

(A) Tetrahedron (4 faces)

(B) Cube (6 faces)

(C) Octahedron (8 faces)

(D) Square pyramid (5 faces)

Question 6. If a solid has 10 vertices, 15 edges, and 7 faces, could it be a convex polyhedron according to Euler's formula?

(A) Yes, $10 - 15 + 7 = 2$.

(B) No, the sum $V - E + F$ is not 2.

(C) Yes, but only if it's concave.

(D) Cannot be determined from Euler's formula alone.