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Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus

Solution:

Given: The diagonals of a quadrilateral bisect each other at right angles.

To show that a given quadrilateral is a rhombus, we have to show it is a parallelogram and all the sides are equal.

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

So, we have, OA = OC, OB = OD, and ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°.

To prove ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD are equal.

In ΔAOD and ΔCOD,

OA = OC (Diagonals bisect each other)

∠AOD = ∠COD = 90° (Given)

OD = OD (Common)

∴ ΔAOD ≅ ΔCOD (By SAS congruence rule)

∴ AD = CD (By CPCT) ……………(1)

Similarly, it can be proved that

AD = AB and CD = BC ……………..(2)

From Equations (1) and (2), AB = BC = CD = AD

Since opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that ABCD is a rhombus.

☛ Check: Class 9 Maths NCERT Solutions Chapter 8

Video Solution:

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 3

Summary:

If the diagonals of a quadrilateral bisect each other at right angles, then we have proved that it is a rhombus.

☛ Related Questions:

- Show that the diagonals of a square are equal and bisect each other at right angles.
- Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
- Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus.
- ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.

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