Question

An isosceles triangle ABC is inscribed in a circle when AB = AC = 20 cm and BC = 24 cm. Find the radius of the circle.

Answer 1

Given:

• The triangle ABC is isosceles with AB = AC = 20 cm and BC = 24 cm.
• We need to find the radius of the circumcircle.

Step 1: Use the formula for the circumradius

The formula for the circumradius R of a triangle is given by:

`R = (abc)/(4A)`

Where:

• a, b, c are the sides of the triangle,
• A is the area of the triangle.

In our case:

• a = 20 cm (side AB),
• b = 20 cm (side AC),
• c = 24 cm (side BC).

We first need to find the area A of the triangle.

Step 2: Find the area of the triangle using Heron's formula

The semi-perimeter s of the triangle is:

`s = (a + b + c)/2`

= `(20 + 20 + 24)/2`

= 32 cm

Now, using Heron’s formula to find the area A:

`A = sqrt(s(s - a)(s - b)(s - c))`

Substitute the values:

`A = sqrt(32(32 - 30)(32 - 20)(32 - 24))`

`A = sqrt(32 xx 12 xx 12 xx 8)`

`A = sqrt(36864)`

A = 192 cm²

Step 3: Calculate the circumradius

Now that we have the area, we can use the circumradius formula:

`R = (abc)/(4A)`

= `(20 xx 20 xx 24)/(4 xx 192)`

`R = 9600/768`

R = 12.5 cm

The radius of the circle is 12.5 cm.