Question

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

• A. 18
• B. 24
• C. \( 48\sqrt{3} \)
• D. \( 144\sqrt{3} \)

Hint:

In problems involving geometric shapes like triangles, use the LCM to find the minimum side length when dividing the sides into equal parts. Then use standard area formulas to calculate the area.

Answer 1

The Correct Option is D

We are given an equilateral triangle PQR where the sides are divided into four, six, and eight equal parts respectively. The length of each subdivided part is an integer, so we need to find the minimum area of the triangle PQR.

Step 1: Determine the side lengths.
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: \( 144\sqrt{3} \)