In the figure, $ A B C $ is a right triangle right-angled at $ B $ such that $ B C=6 \mathrm{~cm} $ and $ A B=8 \mathrm{~cm} $ Find the radius of its incircle."

Given:

In the figure, \( A B C \) is a right triangle right-angled at \( B \) such that \( B C=6 \mathrm{~cm} \) and \( A B=8 \mathrm{~cm} \).

To do:

We have to find the radius of its incircle.

Solution:

In right-angled triangle $ABC$,

$\angle B = 90^o, BC = 6\ cm, AB = 8\ cm$

Let $r$ be the radius of incircle whose centre is $O$ and touches the sides $AB, BC$ and $CA$ at $P, Q$ and $R$ respectively.

$AP$ and $AR$ are the tangents to the circle.

This implies,

$AP = AR$
Similarly,

$CR = CQ$ and $BQ = BP$
$OP$ and $OQ$ are radii of the circle.
$OP\ perp\ AB$ and $OQ\ \perp\ BC$ and $\angle B = 90^o$ 

$BPOQ$ is a square.

$BP = BQ = r$

$AR = AP = AB - BD = 8 - r$

$CR = CQ = BC - BQ = 6 - r$
$AC^2 = AB^2 + BC^2$     (By Pythagoras theorem)

$= 8^2 + 6^2$

$= 64 + 36$

$= 100$

$= (10)^2$
$AC = 10\ cm$

$AR + CR = 10\ cm$

$8 - r + 6 - r = 10$

$14 - 2r = 10$

$2r = 14 - 10$

$2r = 4$

$r = 2\ cm$
The radius of the incircle is 2 cm.

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Updated on: 10-Oct-2022

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