Question Statement:
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.)
Options:
-
(A) 20
7
-
(B) 28
5
-
(C) 9
2
-
(D) 35
4
P
Q
R
S
T
V
7
4
5 90°
90°
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2023 Q.3)**
**1. Governing Concept**:
- **Topic**: Geometry & Mensuration (Parallelogram Area Equivalences)
- In parallelogram $PQRS$, the area can be calculated using base $\times$ corresponding altitude:
$\text{Area} = \text{base}_1 \times \text{height}_1 = \text{base}_2 \times \text{height}_2$
- Given $PS = 7\text{ cm}$, altitude $PT = 4\text{ cm}$ (perpendicular to side $SR$), and altitude $PV = 5\text{ cm}$ (perpendicular to side $QR$):
- In a parallelogram, opposite sides are equal: $QR = PS = 7\text{ cm}$.
- Using base $QR = 7\text{ cm}$ and perpendicular height $PV = 4$ or $5$:
$\text{Area} = QR \times PV = 7 \times 4 = 28\text{ cm}^2 \text{ (or } 7 \times PV\text{)}$
- Since $PT \perp RS$ (or $PV \perp RS$):
$\text{Area} = RS \times PV = QR \times PT \implies RS \times 5 = 7 \times 4 = 28 \implies RS = \frac{28}{5}\text{ cm}$
**2. Option Analysis**:
- **Option (A) [20/7] is INCORRECT**: Inverted ratio.
- **Option (B) [28/5] is CORRECT**: Directly equates parallelogram areas: $RS \times 5 = 7 \times 4 \implies RS = \frac{28}{5}\text{ cm}$.
- **Option (C) [9/2] is INCORRECT**: Arithmetic error.
- **Option (D) [35/4] is INCORRECT**: Incorrect product combination.
**3. Final Answer**:
- **Correct Choice**: **B**
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