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In the fig. ABC is right triangle right angled at B such that BC = 6cm and AB = 8cm. Find the radius of its in circle.

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#### Solution

BC = 6cm AB = 8cm

As ABC is right angled triangle

By Pythagoras theorem

𝐴𝐶^{2} = 𝐴𝐵^{2} + 𝐵𝐶^{2} = 6^{2} + 8^{2} = 100

𝐴𝐶 = 10 𝑐𝑚

Consider BQOP ∠B = 90°,

∠BPO = ∠OQB = 90° [At point of contact, radius is perpendicular to tangent]

All the angles = 90° & adjacent sides are equal

∴ BQOP is square BP = BQ = r

We know that

The tangents drawn from any external point are equal in length.

AP = AR = AB – PB = 8 – r

QC = RC = BC – BQ = 6 – r

AC = AR + RC ⇒ 10 = 8 – r + 6 – r

⇒ 10 = 14 – 2r

⇒ 2r = 4

⇒ Radius = 2cm

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