The sides of a triangle are in the ratio 13 : 14 : 15 and its perimeter is 84 cm. What is the area of the triangle?
336 cm2
Let the sides be 13k, 14k and 15k. Then 13k + 14k + 15k = 42k = 84, so k = 2 and the sides are 26 cm, 28 cm and 30 cm.
By Heron's formula, area = \(\sqrt{s(s - a)(s - b)(s - c)}\), where the semi-perimeter s = 84 ÷ 2 = 42 cm.
So s – a = 42 – 26 = 16, s – b = 42 – 28 = 14 and s – c = 42 – 30 = 12, giving area = \(\sqrt{42 \times 16 \times 14 \times 12} = \sqrt{112896} = 336\) cm2.
Check: a 13–14–15 triangle has area 84 square units, and scaling all sides by k = 2 multiplies the area by 22 = 4, giving 84 × 4 = 336 cm2.
Hence, 336 cm2.
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