Question

In the given figure, PQRS and PXYZ are parallelograms. Prove that they are of equal area.

[Hint: Join XQ. Area (ΔXPQ) = `1/2` of each || gm]

Answer 1

Given:

• Two parallelograms: PQRS and PXYZ
• Hint: Join XQ

To prove: Area (PQRS) = Area (PXYZ) 

Step 1: Use the hint

Join XQ. This will divide both parallelograms into triangles:

• In parallelogram PQRS: Triangle ΔPQX forms half of the parallelogram PXYZ 
• In parallelogram PXYZ: Triangle ΔXPQ also forms half of the parallelogram PXYZ

Step 2: Areas of triangles

Recall the formula for area of a parallelogram:
Area of parallelogram = Base × Height

For triangle ΔXPQ (inside PXYZ):
Area (ΔXPQ) = `1/2` Area (PXYZ)

For triangle ΔXPQ (inside PQRS):
Area (ΔXPQ) = `1/2` Area (PQRS)

Step 3: Equate the areas of the triangles

Since ΔXPQ is common to both parallelograms:

`1/2` Area (PQRS) = `1/2` Area (PXYZ)

Multiply both sides by 2:

Area (PQRS) = Area (PXYZ)