Question

PQR is an isosceles triangle inscribed in a circle. Calculate the radius of the circle if PQ = PR = 50 cm and QR = 60 cm.

Answer 1

We are given an isosceles triangle PQR inscribed in a circle with:

• PQ = PR = 50 cm,
• QR = 60 cm,

And we are asked to find the radius of the circumcircle (the circle in which the triangle is inscribed).

Step 1: Use the formula for the circumradius

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

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

Where:

a,b, c are the sides of the triangle,

A is the area of the triangle.

In our case:

• a = 50 cm (side PQ),
• b = 50 cm (side PR),
• c = 60 cm (side QR).

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`

= `(50 + 50 + 60)/2`

= 80 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(80(80 - 50)(80 - 50)(80 - 60))`

`A = sqrt(80 xx 30 xx 30 xx 20)`

`A = sqrt(1440000)`

A = 1200 cm²

Step 3: Calculate the circumradius

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

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

= `(50 xx 50 xx 60)/(4 xx 1200)`

`R = 150000/4800`

R = 31.25 cm

The radius of the circumcircle is 31.25 cm.