In the figure, a circle touches all the four sides of a quadrilateral $ A B C D $ with $ A B=6 \mathrm{~cm}, B C=7 \mathrm{~cm} $ and $ C D=4 \mathrm{~cm} . $ Find $ A D $."

Given:

A circle touches all the four sides of a quadrilateral \( A B C D \) with \( A B=6 \mathrm{~cm}, B C=7 \mathrm{~cm} \) and \( C D=4 \mathrm{~cm} . \)

To do:

We have to find \( A D \).

Solution:

Let $AD = x$

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

This implies,

$AP = AS$

Similarly,

$BP = BQ$

$CQ = CR$

$DR = DS$

Therefore,

$AP+BP+DR+CR=AS+BQ+DS+CQ$

$AB + CD = AD + BC$

$6 + 4 = 7 + x$

$10 = 7 + x$

$x = 10 - 7$

$x= 3$

Therefore, $AD=3\ cm$.

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Updated on: 10-Oct-2022

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