Question

The base of an isosceles triangle is $8cm$ long and each of its equal sides measures $6cm$. The area of the triangle is

• A. $8\sqrt{5}c{m}^2$
• B. $16\sqrt{5}c{m}^2$
• C. $16\sqrt{3}c{m}^2$
• D. $8\sqrt{3}c{m}^2$

Answer 1

The correct option is A

$8\sqrt{5}c{m}^2$

Explanation for the correct option:

Find the semi-perimeter of the triangle

It is known that in an isosceles triangle, two sides are equal.

So the sides of the triangle will be $a=8cm,b=6cm,c=6cm.$.

It is known that the semi-perimeter of the triangle can be given as: $s=\frac{a+b+c}{2}$.

Where $a,b,c$ are the sides of the triangle.

So the semi-perimeter will be,

$$\begin{gathered} s=\frac{8+6+6}{2} \\ \Rightarrow s=\frac{20}{2} \\ \Rightarrow s=10cm \end{gathered}$$

Find the area of the triangle

Using Heron's formula.

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

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

$$\begin{gathered} A=\sqrt{s(s-a)(s-b)(s-c)} \\ \Rightarrow A=\sqrt{10(10-8)(10-6)(10-6)} \\ \Rightarrow A=\sqrt{10\left(2\right)\left(4\right)\left(4\right)} \\ \Rightarrow A=8\sqrt{5}c{m}^2 \end{gathered}$$

Hence, the correct option is (a) $8\sqrt{5}c{m}^2$.