Chapter 4 Quadrilaterals (Class 8 - Latest Maths NCERT (Ganita Prakash I) Solutions)

Question 1. Find all the other angles inside the following rectangles.

Question 2. Draw a quadrilateral whose diagonals have equal lengths of $8 \text{ cm}$ that bisect each other, and intersect at an angle of

(i) $30^\circ$

(ii) $40^\circ$

(iii) $90^\circ$

(iv) $140^\circ$

Question 3. Consider a circle with centre $O$. Line segments $PL$ and $AM$ are two perpendicular diameters of the circle. What is the figure $APML$? Reason and/or experiment to figure this out.

Question 4. We have seen how to get $90^\circ$ using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact $90^\circ$ using these?

Question 5. We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle?

In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?

Question 1. Find the remaining angles in the following quadrilaterals.

Question 2. Using the diagonal properties, construct a parallelogram whose diagonals are of lengths $7 \text{ cm}$ and $5 \text{ cm}$, and intersect at an angle of $140^\circ$.

Question 3. Using the diagonal properties, construct a rhombus whose diagonals are of lengths $4 \text{ cm}$ and $5 \text{ cm}$.

Question 1. Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides $4 \text{ cm}$.

Question 2. Construct a kite whose diagonals are of lengths $6 \text{ cm}$ and $8 \text{ cm}$.

Question 3. Find the remaining angles in the following trapeziums —

Question 4. Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions —

(i) What is the quadrilateral that is both a kite and a parallelogram?

(ii) Can there be a quadrilateral that is both a kite and a rectangle?

(iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?

Question 5. If $PAIR$ and $RODS$ are two rectangles, find $\angle IOD$.

Question 6. Construct a square with diagonal $6 \text{ cm}$ without using a protractor.

Question 7. $CASE$ is a square. The points $U$, $V$, $W$ and $X$ are the midpoints of the sides of the square. What type of quadrilateral is $UVWX$? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).

Question 8. If a quadrilateral has four equal sides and one angle of $90^\circ$, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.

Question 9. What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.

Hint: Draw a diagonal and check for congruent triangles.

Question 10. Will the sum of the angles in a quadrilateral such as the following one also be $360^\circ$? Find the answer using geometric reasoning as well as by constructing this figure and measuring.

Question 11. State whether the following statements are true or false. Justify your answers.

(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.

(ii) A quadrilateral having three right angles must be a rectangle.

(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.

(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.

(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.

(vi) A quadrilateral in which all the angles are equal is a rectangle.

(vii) Isosceles trapeziums are parallelograms.