Question

Find the area of the following trapezium.

Answer 1

Given trapezium PQRS with

• PQ || SR   ...(Parallel sides)
• PQ = 30 cm
• QR = 29 cm
• SR = 50 cm
• PS perpendicular to SR   ...(Right angle at S)

Stepwise calculation:

1. Draw perpendiculars PM and QN from P and Q to the base SR, intersecting at points M and N respectively.

2. Let PM = QN = height (h) since PQ || SR, these perpendiculars are equal.

3. Let SM = x, so NR = 50 – x + 30 = 20 – x because PQ = MN = 30 cm.

4. Use Pythagoras theorem in triangles PMS and QNR:

h² = PS² – SM²

= 29² – x²

= 841 – x²

h² = QR² – NR²

= 29² – (20 – x)²

= 841 – (400 – 40x + x²)

= 841 – 400 + 40x – x²

= 441 + 40x – x²

5. Equate both expressions for h²:

841 – x² = 441 + 40x – x² 

Simplify:

841 = 441 + 40x

841 – 441 = 40x

400 = 40x

x = 10

6. Now calculate height (h):

`h = sqrt(841 - x^2)`

= `sqrt(841 - 100)`

= `sqrt(741) ≈ 27.22  cm`

7. Calculate area of trapezium:

Area = `1/2` × (PQ + SR) × h

= `1/2 xx (30 + 50) xx 27.22`

= 40 × 27.22

= 1088.8 cm²

This is still off from 840 cm².