State whether the statement is true or false.

The diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

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True

False

answer is A.

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Detailed Solution

Given that, the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Let ABCD be the quadrilateral with equal diagonals bisecting each other. Drawing the figure from the given data.
Question Image

SAS congruence criteria (side-Angle-side): If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.
We have AC = BD. They bisect each other perpendicularly. we get:
OA = OC [1]
OB = OD [2]

∠ A O B = ∠ B O C = ∠ C O D = ∠ A O D = 900

  [3]
In ∆AOD and ∆COD,
OA = OC  [from 1]
OD = OD  [common]

∠ A O D = ∠ C O D

[from 3]
By SAS congruence criteria, we get

Δ A O D ≅ Δ C O D

.
We know that the corresponding parts of congruent triangles are equal.
Thus, by C.P.C.T., we get AD = CD. [4]
Similarly,

Δ B O C ≅ Δ C O D

.
Thus, by C.P.C.T., we get CD = BC. [5]
Similarly,

Δ B O C ≅ Δ A O B

.
Thus, by C.P.C.T., we get AB = BC. [6]
From [4] [5] and [6] we get that AB = BC = CD = DA.
Thus, the quadrilateral with all the sides and the diagonals equal is a square. Hence, ABCD is square, which proves that statement is true.
Therefore, the correct option is 1.

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