Question : The difference between the semi-perimeter and the sides of ΔPQR are 18 cm, 17 cm, and 25 cm, respectively. Find the area of the triangle.
Option 1: $330\sqrt{510}$ cm2
Option 2: $230\sqrt{510}$ cm2
Option 3: $30\sqrt{510}$ cm2
Option 4: $130\sqrt{510}$ cm2
Correct Answer: $30\sqrt{510}$ cm 2
Solution :
Let the semi-perimeter be $s$ and the sides of a triangle are $a,b,$ and $c$.
Given,
$s - a = 18$------- (1)
$s - b = 17$--------(2)
$s - c = 25$--------(3)
⇒ $a = s - 18$
⇒ $b = s - 17$
⇒ $c = s - 25$
Now, $s = \frac{a + b+c}{2}$ = $\frac{s−18+s−17+s−25}{2}=\frac{3s−60}{2}= 60$ units
Now with the help of Heron's formula,
Area of Triangle = $\sqrt{s(s-a)(s-b)(s-c)}$
⇒ Area of triangle = $\sqrt{60×18×17×25}$
⇒ Area of triangle = $\sqrt{459000}$
$\therefore$ Area of triangle = $30\sqrt{510}$ cm$^2$
Hence, the correct answer is $30\sqrt{510}$ cm
2
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