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If the diagonals of a quadrilateral bisect each other at right angles, it will be a
(a) rhombus
(b) trapezium
(c) rectangle
(d) kite

Solution:

Given, the diagonals of a quadrilateral bisect each other at right angles.

We have to find the type of quadrilateral.

If the diagonals of a quadrilateral bisect each other at right angles, it will be a, (a) rhombus, (b) trapezium, (c) rectangle, (d) kite

Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at the right angle.

OA = OC and OB = OD

Now, ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°.

In ΔAOD and ΔCOD,

OA = OC

∠AOD = ∠COD = 90°

OD = OD (Common)

By SAS rule, ΔAOD ≅ ΔCOD

By CPCT,

AD = CD -------------- (1)

Similarly, AD = AB and CD = BC ---------------- (2)

From (1) and (2),

AB = BC = CD = AD

Since opposite sides of quadrilateral ABCD are equal, ABCD is a parallelogram.

Since all sides of a parallelogram ABCD are equal, ABCD is a rhombus.

Therefore, if the diagonals of a quadrilateral bisect each other at right angles, it will be a rhombus.

✦ Try This: If the diagonals of which of the following bisect each other at right angles (a) rhombus, (b) square, (c) both a and b, (d) none of these

☛ Also Check: NCERT Solutions for Class 8 Maths

NCERT Exemplar Class 8 Maths Chapter 5 Solved Problem 3