A window is made up of a square portion and an equilateral triangle portion above it. The base of
the triangular portion coincides with the upper side of the square. If the perimeter of the window is
6 m, the area of the window in m2 is ___________.
Options:
(A) G
0 phase
(B) S phase
(C) G 1 phase
(D) G 2 phase
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2016 Q.05)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Mensuration & Plane Geometry (Composite Perimeter and Area)
**2. Detailed Step-by-Step Derivation**:
- The window is composed of a square of side $s$ and an equilateral triangle of side $s$ placed on top of it.
- The shared boundary between the square and the triangle is internal to the window and does **not** contribute to the outer perimeter.
- The perimeter of the composite window consists of:
- 3 outer edges of the square (bottom, left, and right sides) = $3s$.
- 2 outer edges of the equilateral triangle = $2s$.
- Total perimeter:
$\text{Perimeter} = 3s + 2s = 5s$
- Given that the perimeter is $6\text{ m}$:
$5s = 6 \implies s = \frac{6}{5} = 1.2\text{ m}$
- Now calculate the total area of the window:
- Area of the square portion:
$A_{\text{square}} = s^2 = (1.2)^2 = 1.44\text{ m}^2$
- Area of the equilateral triangle portion:
$A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2 = \frac{1.732}{4} \times 1.44 = 0.433 \times 1.44 \approx 0.6235\text{ m}^2$
- Total area of the window:
$A_{\text{total}} = A_{\text{square}} + A_{\text{triangle}} = 1.44 + 0.6235 = 2.0635\text{ m}^2 \approx 2.06\text{ m}^2$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: 1.43 accounts only for the square portion.
- **Option (B) [CORRECT]**: $2.06\text{ m}^2$ is the exact composite area of the window ($1.44 + 0.6235$).
- **Option (C) [INCORRECT]**: 2.68 assumes a 4-sided square perimeter.
- **Option (D) [INCORRECT]**: 2.88 assumes $s = \sqrt{2}$.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **B**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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