Question

ΔABC circumscribes a circle of radius r such that ∠B = 90°. If AB = 3 cm and BC = 4 cm, then find the value of r.

Answer 1

Here, AB, BC and CA are the tangents to the circle at P, N and M, respectively.

Also, OP = ON = OM = r    ...(Radius of the circle)

Area of ΔABC = `1/2` × BC × AB  

= `1/2 xx 4 xx 3`  ...[∵ Area of Δ = `1/2` × base × height]

= 6 sq. cm

Now, using pythogoras in right ΔABC, we get

CA² = AB² + BC²

⇒ CA² = (3)² + (4)²

⇒ CA² = 9 + 16

⇒ CA² = 25

⇒ CA = 5 cm

Now, Area of ΔABC = Area of ΔOAB + area of ΔOBC + Area of ΔOCA

⇒ 6 = `1/2 xx r xx "AB" + 1/2 xx r xx "BC" + 1/2 xx r xx "CA"`

⇒ 6 = `1/2 xx r("AB" + "BC" + "CA")`

⇒ 12 = r(3 + 4 + 5)

⇒ 12 = 12r

⇒ r = 1 cm