Question

Prove that if the diagonals of a quadrilateral are equal and bisect eachother at right angle then it is a square.

Answer 1

Hey!!!

Let ABCD be a quadrilateral and diagonals intersecting at point O.

Given: In quad ABCD;
OA= OC
OB=OD

To Prove: ABCD is square

Proof: In ∆s AOB and COF
AO= CO (given)
/_ AOB = /_ COD (vertically opposite angles)
OB= OD (given)

Therefo, ∆AOB ≈ ∆COD (SAS)

therefore, AB= CD (CPCT)

AND /_ 1 = /_ 2 (CPCT) ---------- 1)

But from fig. they are alternate angles.

therefore, AB || CD -------- 2)

from 1)& 2)

ABCD is parallelogram --------A)

Now,
In ∆s AOB & COB

OA = OC (given)
/_ AOB = /_ COB (each 90°)
OB = OB (common side)

therefore, ∆AOB≈∆COB (SAS)

therefore, AB=BC (CPCT)

AB=BC=CD=AD --------- 3)

therefore, ABCD is a rhombus. ------B)

In ∆s DAB & CBA

AB=AB (common side)
BD=AC (given)
AD=BC (from 3)

therefore, ∆DAB≈∆CBA (SSS)

Therefore, /_A=/_ B (CPCT)

But they are co-int. angles

therefore, /_A + /_B = 180°

2/_A = 180°

/_ A = 90°

Therefore, ABCD is a rectangle. -----C)

Therefore, from A) B) & C)

ABCD is square.

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