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Chapter 5: Q34 (page 188)

Prove: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Short Answer

Expert verified

A parallelogram is a rhombus if the diagonals of the parallelogram are perpendicular.

Step by step solution

01

Step 1. Consider the diagram.

The figure is made having diagonals AC and BD .

Here, ABCD is a parallelogram.

02

Step 2. State the proof.

In ΔAODand ΔAOB.

OA=OA (Common)

∠AOD≅∠AOB=90∘ (Diagonals are perpendicular)

OD=OB (Diagonals of parallelogram bisects)

So, ΔAOD≅ΔAOB by SAS postulate.

Therefore, by CPCT AD→≅AB→.

Opposite sides are congruent in parallelogram.

AB→≅DC→and AD→≅BC→. (As AB→≅AD→)

That implies, AD→≅DC→≅AB→≅BC→.

Therefore, parallelogram ABCD is a rhombus.

03

Step 3. State the conclusion.

Therefore, parallelogram ABCD is a rhombus if the diagonals of the parallelogram are perpendicular (proved).

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