Question : The ratio of the sides of a triangle is 3 : 3 : 4. If the area of a triangle is $32 \sqrt{5}$ cm2, then what is the length of the equal sides?
Option 1: 12 cm
Option 2: 16 cm
Option 3: 10 cm
Option 4: 15 cm
Correct Answer: 12 cm
Solution :
Let the sides be $3k$, $3k$, and $4k$.
So, semi perimeter, $s = \frac{a+b+c}{2}$ = $\frac{3k+3k+4k}{2}=\frac{10k}{2}={5k}$
Now, $A = \sqrt{s(s - a)(s - b)(s - c)}$
⇒ 32$\sqrt{5}$ = $\sqrt{5k(5k - 3k)(5k - 3k)(5k - 4k)}$
⇒ 32$\sqrt{5}$ = $\sqrt{5k(2k)(2k)( k)}$
⇒ 32$\sqrt{5}$ = $\sqrt{20k^4}$
⇒ (32$\sqrt{5})^2$ = ${20k^4}$
⇒ 1024 × 5 = ${20k^4}$
⇒ 5120 = ${20k^4}$
⇒ ${k^4}$ = 256
⇒ $k$ = 4
Length of equal sides $=3k= 3×4=12$ cm
Therefore, the length of each of the equal sides is 12 cm.
Hence, the correct answer is 12 cm.
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