Question

The sides of a triangle are in the ratio $5:12:13$ and its perimeter is $150cm$. The area of the triangle is

• A. $750c{m}^2$
• B. $375c{m}^2$
• C. $250c{m}^2$
• D. $500c{m}^2$

Answer 1

The correct option is A

$750c{m}^2$

Explanation for the correct option:

Find the sides of the triangle

Let the sides are $a=5x,b=12x,c=13x$ since the ratio is $5:12:13$.

Since it is given that the perimeter is $150cm$.

So,

$$\begin{gathered} 5x+12x+13x=150 \\ \Rightarrow 30x=150 \\ \Rightarrow x=5cm \end{gathered}$$

So, the sides will be

$$\begin{gathered} a=5\times 5=25cm, \\ b=12\times 5=60cm, \\ c=13\times 5=65cm. \end{gathered}$$.

Find the area of the triangle

To find the area of the triangle, we will use Heron's formula.

Heron's formula can be given that: $A=\sqrt{s(s-a)(s-b)(s-c)}$

Where $s=\frac{a+b+c}{2}$ and $a,b,c$ are the sides of the triangle.

Thus, the semi-perimeter, $s=\frac{150}{2}=75cm$.

Hence, the area will be,

$$\begin{gathered} A=\sqrt{75(75-25)(75-60)(75-65)} \\ \Rightarrow A=\sqrt{75\left(50\right)\left(15\right)\left(10\right)} \\ \Rightarrow A=750c{m}^2 \end{gathered}$$

Hence, the correct option is (a) $750c{m}^2$.