Question

An isosceles triangle ABC is inscribed in a circle when AB = AC = 10 cm and BC = 12 cm. Find the radius of the circle.

Answer 1

Step 1:

Draw an altitude AD from A to BC

D bisects BC, so BD = `12/2` = 6 cm

In right triangle ABD, 

Use the Pythagorean theorem:

AD² + BD² = AB²

AD² + 6² = 10²

AD² + 36 = 100

AD² = 64

AD = 8 cm

Step 2:

Let O be the center of the circle and r be the radius

OA = OB = r

OD = AD – OA = 8 – r

In right triangle OBD,

Use the Pythagorean theorem:

OB² = OD² + BD²

r² = (8 – r)² + 6²

Step 3:

r² = 64 – 16r + r² + 36

0 = 100 – 16r

16r = 100

`r = 100/16`

r = `25/4` = 6.25 cm

The radius of the circle is 6.25 cm.