Question

In a triangle ABC, the sides are in the ratio 3 : 4 : 5. If the area of the triangle is 96 sq units, what is its perimeter?

Hint:

Recognizing Pythagorean triplets like (3, 4, 5), (5, 12, 13), (8, 15, 17), etc., can save a lot of time by immediately identifying a triangle as right-angled, which simplifies area calculations.

Answer 1

Correct Answer: 48

Step 1: Let sides be \( 3k, 4k, 5k \). Semi-perimeter \( s=\frac{3k+4k+5k}{2}=6k \).
Step 2: By Heron's formula, Area \( =\sqrt{s(s-3k)(s-4k)(s-5k)}=\sqrt{6k\cdot3k\cdot2k\cdot k}=\sqrt{36k^4}=6k^2 \).
Step 3: Setting \( 6k^2=96 \) gives \( k^2=16 \), so \( k=4 \), and the sides become \( 12,16,20 \).
\[ \boxed{\text{Perimeter}=12+16+20=48} \]