The sides of a triangle are in the ratio 8 : 9 : 10 and its perimeter is 270 cm. What is its area?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(225\sqrt{231}\,\mathrm{cm^2}\)
When all three sides of a triangle are known, its area comes from Heron’s formula: Area \(=\sqrt{s(s-a)(s-b)(s-c)}\), where \(s\) is half the perimeter.
The sides (ratio 8 : 9 : 10, perimeter 270 cm):
\(8k+9k+10k=270\), so \(k=10\)
Sides: 80 cm, 90 cm and 100 cm
Heron’s formula:
\(\displaystyle s=\frac{270}{2}=135\)
\(s-a=55\), \(s-b=45\), \(s-c=35\)
Area \(=\sqrt{135\times55\times45\times35}\)
\(=\sqrt{3^5\times5^4\times7\times11}\)
\(=3^2\times5^2\times\sqrt{3\times7\times11}=225\sqrt{231}\,\mathrm{cm^2}\)