GATE 2021 AE – Question 10
In equilateral triangle PQR, PQ is divided into four equal parts, QR into six and PR into eight. Each part has integer length in cm. The minimum area in cm² is
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Correct answer: (D) $144\sqrt3$
Explanation
The sides of the equilateral triangle PQR are all equal, say $s$ cm.
- PQ is divided into 4 equal parts, each of integer length, so $4\mid s$.
- QR is divided into 6 equal parts of integer length, so $6\mid s$.
- PR is divided into 8 equal parts of integer length, so $8\mid s$.
The smallest side satisfying all three is the least common multiple:
$$\text{lcm}(4,6,8)=24\text{ cm}.$$
**Area of an equilateral triangle:**
$$A=\frac{\sqrt3}{4}s^2=\frac{\sqrt3}{4}(24)^2=\frac{\sqrt3}{4}\times576=\mathbf{144\sqrt3\ cm^2}.$$