Question

Find the area of a trapezium whose parallel sides are 11 cm and 25 cm long and nonparallel sides are 15 cm and 13 cm.

Answer 1

Given: Parallel sides (bases) = 25 cm and 11 cm; nonparallel sides (legs) = 15 cm and 13 cm.

Step-wise calculation:

1. Let the longer base AB = 25, shorter base CD = 11.

Drop perpendiculars from C and D to AB, meeting AB at E and F respectively.

Let CE = DF = h (height) and let AE = x.

Then EF = CD = 11.

So BF = AB – AE – EF 

= 25 – x – 11 

= 14 – x

2. In the right triangles with the legs:

For the side of length 15:

x² + h² = 15² 

= 225

For the side of length 13:

(14 – x)² + h²

= 13² 

= 169

3. Subtract the second equation from the first:

x² – (14 – x)² = 225 – 169

= 56

Expand: x² – (196 – 28x + x²) = 56 

⇒ 28x – 196 = 56

⇒ 28x = 252

⇒ x = 9

4. Substitute x = 9 into x² + h² = 225:

9² + h² = 225

⇒ 81 + h² = 225

⇒ h² = 144

⇒ h = 12 cm

5. Area of trapezium = `1/2 xx ("sum of parallel sides") xx "height"` 

= `1/2 xx (25 + 11) xx 12`

= `1/2 xx 36 xx 12`

= 216 cm²

The area of the trapezium is 216 cm².