NCERT Solutions for Class 9 Maths Chapter 8

Important NCERT Questions

Quadrilaterals

NCERT Books for Session 2022-2023

CBSE Board and UP Board Others state Board

EXERCISE 8.1

Page No:146

Questions No: 5

# Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

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Given: ABCD is a quadrilateral such that AC = BD, AO = CO, BO = DO and ∠COD = 90°.

To prove: ABCD is a square.

Solution: If the diagonals of a quadrilateral bisects each other at right angle, it is a rhombus .

Hence, AB = BC = CD = DA

In ΔBAD and ΔABC,

AD = BC [∵ Proved above]

BD = AC [∵ Given]

AB = AB [∵ Common]

Hence, ΔBAD ΔABC [∵ SSS Congruency rule]

∠BAD = ∠ABC [∵ CPCT]

But, ∠BAD + ∠ABC = 180° [∵ Co-interior angles]

⇒ 2∠ABC = 180° [∵ ∠BAD = ∠ABC]

⇒ ∠ABC = (180°/2) = 90°

Hence, if the diagonals of a quadrilateral are equal and bisect each other at right angles, than it is a square.