Question

Find the area of an isosceles triangle each of whose equal sides measures 12 cm and whose base is 18 cm.​

Answer 1

• The area of an isosceles triangle = 27√7 cm².

Given :

The sides of an isosceles triangle are :

The equal sides of an isosceles triangle is 12 cm.

• The first angle = 12 cm.
• The second side = 12 cm.
• The third side = 18 cm.

To Find :

• The area of an isosceles triangle.

Solution :

By heron's formula,

$\blue{ \huge{ \star}} \: \: \: \large \boxed{ \tt\pink{ \sqrt{s(s - a)(s - b)(s - c)} }}$

Where,

• s = semi perimeter.
• a = the first side.
• b = the second side.
• c = the third side.

We know that,

Semi perimeter = half of the perimeter.

Semi perimeter = $\sf \dfrac{Perimeter}{2}$

We know that,

Perimeter of the triangle = a + b + c

That means,

Semi perimeter = $\sf \dfrac{a + b + c}{2}$

We have,

• a = 12 cm.
• b = 12 cm.
• c = 18 cm.

$: \implies \rm s = \dfrac{12 + 12 + 18}{2} \: \: cm \\ \\ : \implies \rm s = \dfrac{42}{2} \: \: cm \\ \\ : \implies \rm s = 21 \: \: cm \\ \\ \: \: \rm \therefore \: \: s = 21 \: cm$

Now, substitute all the values (values of s, a, b and c) in the heron's formula.

$: \implies \rm \sqrt{21 \: cm(21 \: cm - 12 \: cm)(21 \: cm - 12 \: cm)(21 \: cm - 18 \: cm)} \\ \\ : \implies \rm \sqrt{21 \: cm(9 \: cm)(9 \: cm)(3 \: cm)} \\ \\ : \implies \rm \sqrt{(3 \times 7)(3 \times 3)(3 \times 3)(3) \: {cm}^{4} } \\ \\ : \implies \rm \sqrt{3 \times 7 \times 3 \times 3 \times 3 \times 3 \times 3 \: {cm}^{4} } \\ \\ : \implies \rm \sqrt{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 7 \: {cm}^{4} } \\ \\ : \implies \rm \sqrt{ {(3)}^{2} \times {(3)}^{2} \times {(3)}^{2} \times 7 \: {cm}^{4} } \\ \\ : \implies \rm 3 \times 3 \times 3 \sqrt{7} \: {cm}^{2} \\ \\ : \implies \rm 9 \times 3 \sqrt{7} \: {cm}^{2} \\ \\ : \implies \rm 27 \sqrt{7} \: {cm}^{2}$

Hence,

The area of an isosceles triangle is 27√7 cm².