# Find the Radius of the Circle and Area of the Triangle Inside the Circle

## What is the area of the triangle inside the circle and radius of the circle?

From the figure, AB and CD are chords of the circle and both chords are perpendicular to each other, then find the radius of the circle and area of the blue triangle if o is the centre of the circle

## Solution: Radius of the circle and area of the triangle

To find the area of the triangle we need to find the radius of the circle

From the figure

## Apply the chord intersection theorem

MA × MB = MC × MD

⇒ 8 × 12 = 6 × MD

⇒ MD = 16 cm

Connect OB and OP then we can form the triangle POB

Let radius of the circle is r

From figure

CD = MC + MD = 6 + 16 = 22 cm

AB = MA + MB = 8 + 12 = 20 cm

Apply Pythagorean theorem in triangle POB

OB² = OP² + BP²

Where, OP = QM = CD – (MC + QD)

QD = CD/2 = 22/2 = 11 cm

Now, OP = 22 – (6 + 11) = 5 cm

BP = AB/2 = 20/2 = 10 cm

Then, OB² = 5² + 10²

⇒ r² = 25 + 100

so, r² = 125

⇒ r = 5√5 cm

so, **radius of the circle = 5√5 cm**

We know MD = 16 cm, then we can find BD from the figure using the Pythagorean theorem

Apply Pythagorean theorem in triangle BMD

BD² = MB² + MD²

⇒ BD² = 12² + 16²

⇒ BD² = 144 + 256

then, BD² = 400

⇒ BD = 20 cm

Connect OF

Apply Pythagorean theorem in triangle OFB

OF² = BO² + BF²

⇒ OF² = (5**√**5)² – 10²

⇒ OF² = 125 – 100

thus, OF² = 25

⇒ BD = 5 cm

Now area of the triangle = ½ × 20 × 5

⇒ **Area of the triangle = 50 cm²**