Question

A lateral side of an isosceles triangle is 15 cm and the altitude is 8 cm. What is the radius of the circumscribed circle

Answer 1

Answer : -

Given that,

Length of lateral side of an isosceles triangle = 15 cm

Altitude = 8 cm

To find,

The radius of circumscribed circle.

Assumption : -

Let us assume that the circumscribed circle is inside the isosceles triangle.

We know that,

The circumcentre, incentre, or the other centre and the centroid are collinear in a isosceles triangle.

In the above figure,

In △ABC, segment AD is a altitude.

AD will bisect BC. [the line perpendicular through the centre of the circle bisects the chord (line)]

Therefore, BD = DC

It's given that, BC is 15 cm.

So, BD = DC = $\frac{15}{2}$ cm

Here, I've taken 'O' as the centre of the circle. As when we look closer we'll observe that, △BOD is a right - angled triangle, where the right angle is on D.

So, △BOD is a right - angled triangle

As, AD is the altitude,

Segment OA = $\frac{2}{3}$ AD

=> $\frac{2}{3}$ 8 cm

=> $\frac{16}{3}$ cm

Segment OD = $\frac{1}{3}$ AD

=> $\frac{1}{3}$ 8 cm

=> $\frac{8}{3}$ cm

Ratio of $\frac{16}{3} and \frac{8}{3}$

=> Ratio = 2 : 1

Here, the centroid of the triangle divided the altitude in the ratio of 2 : 1.

As in the above figure,

Segment OD is the radius of the circumscribed circle.

Therefore, segment OD = $\frac{8}{3}$ cm [As solved above]

So, segment OD = 2.6 cm

Therefore, the radius of the circle is 2.6 cm.

Please refer to the above attachment for better understandings.