If the perimeter of a right-angled triangle is 30~\text{cm} and the hypotenuse is 13~\text{cm}, then what is the area of the triangle?
- A. 24~\text{cm}^2
- B. 27~\text{cm}^2
- C. 30~\text{cm}^2 ✓
- D. 36~\text{cm}^2
Correct Answer: C. 30~\text{cm}^2
Explanation
Let the legs be a and b. We know a+b+13 = 30, so a+b=17. By Pythagoras, a^2+b^2=13^2=169. Squaring the first equation: (a+b)^2 = a^2+b^2+2ab \implies 289 = 169 + 2ab \implies 2ab = 120. The area is \frac{1}{2}ab = 30~\text{cm}^2.
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