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.