In an equilateral triangle PQR, side QR is divided into four equal parts, side PR is divided into six equal parts and si...

Question

In an equilateral triangle PQR, side QR is divided into four equal parts, side PR is divided into six equal parts and side PQ 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 cm^2, is

A. 18
B. 24
C. 48sqrt(3)
D. 144sqrt(3)

Accepted Answer

Minimum area = 144*sqrt(3) cm^2 The side QR is divided into 4 equal parts, PR into 6 equal parts, and PQ into 8 equal parts. For each subdivided length to be an integer, the full side length must be divisible by 4, by 6, and by 8 simultaneously. We need the smallest such integer side length. Since all three sides of an equilateral triangle are equal, the side length must be a common multiple of 4, 6, and 8. The minimum is LCM(4, 6, 8) = 24 cm. The area of an equilateral triangle with side a is (sqrt(3)/4) * a^2.