Question

Find the perimeter and area of trapezium PQRS.

Answer 1

Given that the trapezium PQRS has the dimensions:

• PQ = 4 cm
• QR = 5 cm   ...(Perpendicular height)
• SR = 16 cm

Stepwise calculation:

1. The segment PS is not calculated by diagonal length from the triangle PQS. Instead, recognize that trapezium PQRS contains a rectangle QRPQ (4 cm by 5 cm) and a right triangle formed by PS, SR and the leftover horizontal segment.

2. From the figure, since QR = 5 cm is perpendicular and PQ = 4 cm is horizontal, the total length along the base SR (16 cm) minus the base PQ (4 cm) gives the length of the bottom right segment adjacent to PS as (16 – 4) = 12 cm.

3. Given PS is perpendicular to SR common assumption or can be inferred, then PS is the hypotenuse of a right triangle with sides QR = 5 cm and the difference segment = 12 cm.

4. Calculate PS:

`PS = sqrt(QR^2 + (SR - PQ)^2)`

= `sqrt(5^2 + 12^2)`

= `sqrt(25 + 144)`

= `sqrt(169)`

= 13 cm

5. Now calculate the perimeter by summing all sides:

Perimeter = PQ + QR + SR + PS

= 4 + 5 + 16 + 13

= 38 cm

6. The area is calculated as the average length of the parallel sides multiplied by the height QR:

Area = `1/2` × PQ + SR × QR

= `1/2` × 4 + 16 × 5` 

= `1/2` × 20 × 5` 

= 50 cm²

Perimeter of trapezium PQRS = 38 cm.

Area of trapezium PQRS = 50 cm².