Quadrilaterals — Class 9 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Quadrilaterals" — 8 important questions with detailed answers for CBSE board exam preparation.

TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Quadrilaterals" — 8 important questions with detailed answers for CBSE board exam p…

By Syllab.in · Updated Jun 14, 2026

Key Questions Covered:

State the midpoint theorem for quadrilaterals: In quadrilateral PQRS, if L, M…
Define and differentiate between a parallelogram and a trapezium. If ABCD is …
In rectangle PQRS, the diagonals PR and QS intersect at O. If PQ = 6 cm and Q…
In rhombus ABCD, the diagonals AC and BD intersect at O. If AC = 12 cm and BD…
In trapezium ABCD, AB ∥ CD, AB = 10 cm, CD = 6 cm, and the perpendicular dist…
In square PQRS with side length 5 cm, find the length of the diagonal and ver…
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
State the midpoint theorem for quadrilaterals: In quadril…	✓ Solved
Define and differentiate between a parallelogram and a tr…	✓ Solved
In rectangle PQRS, the diagonals PR and QS intersect at O…	✓ Solved
In rhombus ABCD, the diagonals AC and BD intersect at O. …	✓ Solved
In trapezium ABCD, AB ∥ CD, AB = 10 cm, CD = 6 cm, and th…	✓ Solved
In square PQRS with side length 5 cm, find the length of …	✓ Solved

Showing 6 of 8 questions

Q1: State the midpoint theorem for quadrilaterals: In quadrilateral PQRS, if L, M, N, O are the midpoints of sides PQ, QR, RS, SP respectively, prove that LMNO is a parallelogram.

Step 1: Midpoint Theorem for Quadrilaterals: The figure formed by joining the midpoints of the sides of any quadrilateral is a parallelogram. Step 2: Let PQRS be a quadrilateral with: L = midpoint of PQ M = midpoint of QR N = midpoint of RS O = midpoint of SP Step 3: In triangle PQR, L and M are midpoints of PQ and QR respectively. By midpoint theorem in triangle: LM ∥ PR and LM = PR/2 Step 4: In triangle RPS, N and O are midpoints of RS and SP respectively. By midpoint theorem in triangle: N...

Q2: Define and differentiate between a parallelogram and a trapezium. If ABCD is a parallelogram with AB = 8 cm and perpendicular distance between AB and CD is 5 cm, find the area.

Step 1: Parallelogram: A quadrilateral with both pairs of opposite sides parallel and equal. Properties: Opposite sides equal, opposite angles equal, diagonals bisect each other. Step 2: Trapezium: A quadrilateral with only ONE pair of parallel sides. Properties: One pair of parallel sides, other sides (legs) are not necessarily equal. Step 3: Key Difference: Parallelogram has TWO pairs of parallel sides; trapezium has only ONE pair. Step 4: For the given parallelogram ABCD: AB = 8 cm (base) ...

Q3: In rectangle PQRS, the diagonals PR and QS intersect at O. If PQ = 6 cm and QR = 8 cm, find the length of diagonal PR and verify that O divides both diagonals into equal parts.

Step 1: In rectangle PQRS: PQ = 6 cm (length) QR = 8 cm (breadth) All angles are 90° Step 2: Triangle PQR is a right triangle with right angle at Q. Step 3: By Pythagoras theorem: PR² = PQ² + QR² PR² = 6² + 8² PR² = 36 + 64 PR² = 100 PR = 10 cm Step 4: In a rectangle, both diagonals are equal and bisect each other. Therefore, QS = PR = 10 cm Step 5: Since diagonals bisect each other at O: PO = PR/2 = 10/2 = 5 cm OR = PR/2 = 10/2 = 5 cm So PO = OR ✓ Step 6: Similarly: QO = QS/2 = 10/2 = 5 cm...

Q4: In rhombus ABCD, the diagonals AC and BD intersect at O. If AC = 12 cm and BD = 16 cm, find the side length of the rhombus.

Step 1: Properties of rhombus: - All sides are equal - Diagonals bisect each other at right angles (90°) - Diagonals divide the rhombus into four congruent right triangles Step 2: Given: AC = 12 cm (first diagonal) BD = 16 cm (second diagonal) Step 3: Since diagonals bisect each other: AO = AC/2 = 12/2 = 6 cm BO = BD/2 = 16/2 = 8 cm Step 4: In right triangle AOB (with right angle at O): AB² = AO² + BO² AB² = 6² + 8² AB² = 36 + 64 AB² = 100 AB = 10 cm Step 5: Since all sides of a rhombus are ...

Q5: In trapezium ABCD, AB ∥ CD, AB = 10 cm, CD = 6 cm, and the perpendicular distance between the parallel sides is 4 cm. Calculate the area of the trapezium.

Step 1: Trapezium ABCD has one pair of parallel sides: AB = 10 cm (one parallel side) CD = 6 cm (other parallel side) Perpendicular distance = 4 cm (height) Step 2: Formula for area of trapezium: Area = (1/2) × (sum of parallel sides) × height Area = (1/2) × (a + b) × h Where a and b are the parallel sides, and h is the perpendicular distance. Step 3: Substitute values: Area = (1/2) × (AB + CD) × height Area = (1/2) × (10 + 6) × 4 Area = (1/2) × 16 × 4 Area = (1/2) × 64 Area = 32 cm² Step 4:...

Q6: In square PQRS with side length 5 cm, find the length of the diagonal and verify using the Pythagoras theorem.

Step 1: Square PQRS with side length = 5 cm All sides are equal: PQ = QR = RS = SP = 5 cm All angles are 90° Step 2: Consider triangle PQR formed by two adjacent sides and a diagonal: angle Q = 90° PQ = 5 cm QR = 5 cm Step 3: By Pythagoras theorem: PR² = PQ² + QR² PR² = 5² + 5² PR² = 25 + 25 PR² = 50 PR = √50 PR = √(25 × 2) PR = 5√2 cm Step 4: Alternative formula for diagonal of square: For a square with side a, diagonal = a√2 Diagonal = 5√2 cm ✓ Step 5: Numerical value: 5√2 ≈ 5 × 1.414 5√2 ...

Showing 6 of 8 questions. Visit the full page for complete solutions.