Q.3 In the given figure, PQRS is a parallelogram with PS = 7 cm, PT = 4 cm and
PV = 5 cm. What is the length of RS in cm? (The diagram is representative.)
(A) 20/7
(B) 28/5
(C) 9/2
(D) 35/4
PQRS Parallelogram PS 7 cm PT 4 cm PV 5 cm RS Length Solved
PQRS is a parallelogram where PS equals 7 cm, with perpendiculars PT=4 cm to side QR and PV=5 cm to side RS. The goal is to find RS length among options (A) 20/7, (B) 28/5, (C) 9/2, (D) 35/4.
🔍 Quick Solution
Area = QR × PT = PS × PT = 7 × 4 = 28 cm²
RS = Area ÷ PV = 28 ÷ 5 = 28/5 cm → Answer: (B)
📐 Diagram Setup
In parallelogram PQRS:
- PS ∥ QR and PS = QR = 7 cm (opposite sides equal)
- PQ ∥ SR and PQ = SR = RS (length to find)
- PT = 4 cm (⊥ distance from P to line QR)
- PV = 5 cm (⊥ distance from P to line SR)
📊 Area Method (Key Concept)
Area of parallelogram = base × corresponding height:
Method 1: Base QR = 7 cm, Height PT = 4 cm
Area = 7 × 4 = 28 cm²
Method 2: Base SR = RS (unknown), Height PV = 5 cm
Area = RS × 5 = 28 cm²
∴ RS = 28/5 cm
✅ Option Analysis
Area check: 28/5 × 5 = 28 cm² ✓ Matches perfectly
Area check: 20/7 × 5 ≈ 14.29 cm² ≠ 28
Area check: 4.5 × 5 = 22.5 cm² ≠ 28
Area check: 8.75 × 5 = 43.75 cm² ≠ 28
🎯 GATE Exam Tip
This question tests three key parallelogram properties:
- Opposite sides equal: PS = QR = 7 cm
- Area = base × height (multiple valid pairs)
- Perpendicular distances from vertex to opposite sides
🚀 One-Line Solution
RS = (PS × PT) / PV = (7 × 4) / 5 = 28/5 cm
