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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