Question : The sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 90 cm. Find its area (in cm2).
Option 1: 150
Option 2: 270
Option 3: 30
Option 4: 60
Correct Answer: 270
Solution :
Let the sides of the triangle be $5x, 12x,$ and $13x$ (where $x$ is a positive constant)
The perimeter of the triangle is the sum of its sides:
⇒ $5x+12x+13x=90$
⇒ $30x = 90$
⇒ $x = \frac{90}{30}$ = 3
first side = $5x$ = 5 × 3 = 15 cm
second side = $12x$ = 12 × 3 = 36 cm
third side = $13x$ = 13 × 3 = 39 cm
The semi-perimeter is, $s = \frac{15 + 36 + 39}{2}$ = 45
⇒ Area = $\sqrt{s × (s − a) × (s − b) × (s − c)}$
= $\sqrt{45 × (45 − 15) × (45 − 36) × (45 − 39)}$
= $\sqrt{45 × 30 × 9 × 6}$
= $\sqrt{72900}$
= 270 cm
2
Hence, the correct answer is 270.
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