In trapezium PQRS, PQ || SR and the ratio of PQ to SR is 3:2. If the area of…
2021
In trapezium PQRS, PQ || SR and the ratio of PQ to SR is 3:2. If the area of the trapezium is 480 cm² and the distance between PQ and SR is 12 cm, then the length of SR is
Answer: B. 32 cm — Concept A trapezium has exactly one pair of parallel sides, and its area depends only on those two parallel sides and the perpendicular distance between them:…
- A.
24 cm
- B.
32 cm
- C.
36 cm
- D.
48 cm
Show answer & explanation
Correct answer: B
Concept
A trapezium has exactly one pair of parallel sides, and its area depends only on those two parallel sides and the perpendicular distance between them:
Area = 1/2 × (sum of the parallel sides) × perpendicular height
When the two parallel sides are given as a ratio rather than as lengths, write them as multiples of one unknown. A single equation in that unknown then fixes both lengths at once.
Application
Let x be the common multiplier. Since PQ : SR = 3 : 2, write PQ = 3x and SR = 2x.
Substitute the given area and height into the formula: 480 = 1/2 × (3x + 2x) × 12.
Add the parallel sides first: 3x + 2x = 5x, so 480 = 1/2 × 5x × 12.
Simplify the right-hand side: 1/2 × 12 = 6, and 6 × 5x = 30x, so 480 = 30x.
Divide both sides by 30: x = 480 ÷ 30 = 16.
The question asks for SR, the smaller multiple: SR = 2x = 2 × 16 = 32 cm.
Cross-check
Rebuild both parallel sides from x = 16 and recompute the area independently:
Quantity | Value |
|---|---|
PQ = 3x | 48 cm |
SR = 2x | 32 cm |
PQ : SR | 48 : 32 = 3 : 2 |
Area = 1/2 × (48 + 32) × 12 | 480 cm² |
Both the ratio and the area come out exactly as stated, so SR = 32 cm.
Watch out
The 3 : 2 ratio is written PQ first, so 3x belongs to PQ. Solving correctly but then quoting 3x gives 48 cm, which is PQ — the longer parallel side — not the SR the question asks for.