Question

Area of the triangle whose sides are $13\mathrm{cm},9\mathrm{cm}$ and $6\mathrm{cm}$ is:

• A. $23.6{c\mathrm{m}}^2$
• B. $26.3{c\mathrm{m}}^2$
• C. $36.34{c\mathrm{m}}^2$
• D. $23.66{c\mathrm{m}}^2$

Answer 1

The correct option is D

$23.66{c\mathrm{m}}^2$

Step 1: Given data:

Sides of triangle $a=13\mathrm{cm},b=9\mathrm{cm}$ and $c=6\mathrm{cm}$.

Step 2: Heron's Formula:

Area of the triangle having sides $a,b$ and $c$ can be determined by using the formula:

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

Where, Semi-perimeter $s=\frac{a+b+c}{2}$

Step 3 Finding the semi-perimeter of the triangle:
The sides of the triangle are $a=13cm,b=9cm,c=6cm$
So, the value of the semi-perimeter of the triangle will be:
$$\begin{gathered} \therefore s=\frac{a+b+c}{2} \\ =\frac{13+9+6}{2} \\ =\frac{28}{2} \\ =14cm \end{gathered}$$

Step 4 Find the area of triangle:
Area of triangle will be:
$$\begin{gathered} =\sqrt{s(s-a)(s-b)(s-c)} \\ A=\sqrt{14(14-13)(14-9)(14-6)} \\ =\sqrt{14\times 1\times 5\times 8} \\ =\sqrt{2\times 7\times 5\times 2\times 2\times 2} \\ =4\sqrt{35} \\ =23.66c{m}^2 \end{gathered}$$

Hence, the area of the triangle will be $=23.66{c\mathrm{m}}^2$