Question : What is the area of a triangle having a perimeter of 32 cm, one side of 11 cm, and the difference between the other two sides is 5 cm?
Option 1: $8\sqrt{30}$ cm2
Option 2: $5\sqrt{35}$ cm2
Option 3: $6\sqrt{30}$ cm2
Option 4: $8\sqrt{2}$ cm2
Correct Answer: $8\sqrt{30}$ cm 2
Solution : Let the sides of the triangle be $a$, $b$, and $c$. Given Perimeter = 32 cm ⇒ $a+b+c=32$ ...... equation (1) One side, let $a$ =11 cm Also, the difference between the other two sides, $b−c=5$........ equation (2) Performing equation (1) + equation (2) $a+2b=37$ ⇒ $11+2b=37$ ⇒ $2b=26$ ⇒ $b=13$ cm Now, using equation (2), $c=b−5$ ⇒ $c=13−5=8$ cm Area of the triangle according to Heron's formula = $\sqrt{s(s−a)(s−b)(s−c)}$ where $s$ is the semi-perimeter $s=\frac{11+13+8}{2} = 16$ cm Area $= \sqrt{16(16−11)(16−13)(16−8)}$ $= \sqrt{16(5)(3)(8)}$ $= 8\sqrt{30}$ cm 2 Hence, the correct answer is $8\sqrt{30}$ cm 2 .
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