Find the Radius of the Circle Inside a Right Triangle
Geometry math problem: When two equal circles are inscribed inside a right triangle, which is tangential to two sides of the triangle then find the radius of the circles

ABC is a right angle triangle, AB = 12 cm and AC = 9 cm. find the radius of the circle when the radius of the circles are equal
Solution: Radius of the circle
We can solve this problem in different ways, one of the methods is shown here
This figure can be redrawn as shown below.

Let assume Radius of the circle = r and BQ = BR = x {BQ & BR are tangent of the circle from point B}
AB = AP + PQ + QB = r + 2r + x = 3r + x
⇒ 3r + x = 12……………………………………….eq(1)

From above figure
Let ∠QBR = θ then ∠QOR = 180 – θ
Apply Pythagorean theorem in triangle ABC
BC2 = AB2 + AC2 = 122 + 92 = 144 + 81 = 225
⇒ BC = 15 cm
cos θ = AB / BC = 12 / 15 = 4/5
From triangle BQR
QR2 = BQ2 + BR2 – 2 × BQ × BR × cos θ
⇒ QR2 = x2 + x2 – 2 × x × x × 4/5
⇒ QR2 = 2x2 – 8x2/5
⇒ QR2 = 2x2/5
From triangle QBR
QR2 = OQ2 + OR2 – 2 × OQ × OR × cos (180-θ)
⇒ QR2 = r2 + r2 + 2 × r × r × 4/5 {cos (180-θ) = -cos θ}
⇒ QR2 = 2r2 + 8r2/5
⇒ QR2 = 18r2/5
⇒ QR2 = 2x2/5 = 18r2/5
⇒ x2 = 9r2
⇒ x = 3r
From equation 1
3r + x = 12 = 3r + 3r = 6r
⇒ r = 2 cm