In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equal parts. The length of each subdivided part in cm is an integer.
The minimum area of the triangle PQR possible, in cm², is
Show Hint
- 18
- 24
- \( 48\sqrt{3} \)
- \( 144\sqrt{3} \)
The Correct Option is D
Solution and Explanation
Let the side length of the equilateral triangle PQR be \( s \). The number of parts that each side is divided into is:
- \( PQ \) is divided into 4 parts.
- \( QR \) is divided into 6 parts.
- \( PR \) is divided into 8 parts.
Since the length of each subdivided part must be an integer, the least common multiple (LCM) of 4, 6, and 8 will give the minimum integer side length that satisfies the conditions. The LCM of 4, 6, and 8 is 24. Therefore, the minimum side length of the triangle is \( s = 24 \). Step 2: Calculate the area of the equilateral triangle.
The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula: \[ A = \frac{s^2 \sqrt{3}}{4} \] Substitute \( s = 24 \): \[ A = \frac{24^2 \sqrt{3}}{4} = \frac{576 \sqrt{3}}{4} = 144 \sqrt{3} \] Thus, the minimum area of the triangle PQR is \( 144\sqrt{3} \) cm², which corresponds to option (D). Final Answer: \( 144\sqrt{3} \)
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