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