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Chapter 5: Q34 (page 194)

Prove that if the diagonals of parallelograms are perpendicular, then the parallelogram is a rhombus.

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Check the figure.

Consider the figure.

02

Step 2. Step description.

The CPCTC theorem states that when two triangles are congruent, then every corresponding part of one triangle is congruent to the other.

03

Step 3. Step description.

From the figure, ΔAODand ΔAOB.

OA=OA (Sides are common)

As the diagonals are perpendicular such that ∠AOB≅∠AOD=90∘and the diagonals of a parallelogram bisect such that OB=OD.

Therefore, by the side-angle-side postulate ΔBOA≅ΔBOA.

Now, by CPCT theorem AD→≅AB→.

As the opposite sides are congruent in parallelogram thus

AB→≅DC→&AD→≅BC→         ∴AB→≅AD→⇒AD→≅DC→≅AB→≅BC→ 

Therefore, the parallelogram is a rhombus.

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