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: 4

# Show that the diagonals of a square are equal and bisect each other at right angles.

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Given: ABCD is a square.

To prove: AC = BD, AO = CO, BO = DO and ∠COD = 90°.

Solution: ΔBAD and ΔABC,

AD = BC [∵ Opposite sides of a square]

∠BAD = ∠ABC [∵ Each 90°]

AB = AB [∵ Common]

Hence, ΔBAD ≅ ΔABC [∵ SAS Congruency rule]

BD = AC [∵ CPCT]

In ΔAOB and ΔCOD,

∠OAB = ∠OCD [∵ Alternate angles]

AB = CD [∵ Opposite sides of a square]

∠OBA = ∠ODC [∵ Alternate angles]

Hence, ΔBAD ≅ ΔABC [∵ ASA Congruency rule]

AO = OC, BO = OD, [∵ CPCT]

In ΔAOB and ΔAOD.

OB = OD [∵ Proved above]

AB = AD [∵ Sides of a square]

OA = OA [∵ Common]

Hence, ΔBAD ≅ ΔABC [∵ SSS Congruency rule]

∠AOB = ∠AOD [∵ CPCT]

But. ∠AOB + ∠AOD = 180° [∵ Linear pair]

⇒ 2∠AOB = 180° [∵ ∠AOD = ∠AOB]

⇒ ∠AOB = (180/2) = 90°

Hence, the diagonals of a square are equal and bisect each at right angles.