Question

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

Answer 1

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Answer 2

Solution :

Given,

• AC = BD
• OA = OB and OB = OD
• ∠1 = ∠2 = ∠3 = ∠4 = 90°

To prove,

• ABCD is a square.

Proof,

In ∆AOB and ∆DOC

➪ OA = OC [Given]

➪ OB = OD [Given]

➪ ∠1 = ∠2 [90°]

.°. ∆AOB ≅ DOC. (SAS)

AB = CD [C.P.C.T]

∠5 = ∠6. [ C.P.C.T]

AB || CD

Then,

In ∆AOB and ∆BOC

➪ OA = OC [Given]

➪ OB = OB [Common]

➪ ∠1 = ∠2 [90°]

.°. ∆AOB ≅ ∆BOC (SAS)

AB = BC (C.P.C.T)

Now,

In ∆ABD and ∆ABC

➪ AB = AB (common)

➪ AC = BD (Given)

➪ AD = BC (Side of square)

.°. ∆ABD ≅ ∆ABC. (SSS)

∠A = ∠B. (C.P.C.T)

➪ ∠A + ∠B = 180°. (Co-interior angles)

➪ ∠A + ∠A = 180°

➪ 2∠A = 180°

➪ ∠A = 90°

.°. ∠B = 90°

Similarly, ∠B = ∠C = ∠D = 90°

Hence, ABCD is a square.