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In figure below, $AE$ is the bisector of the exterior $∠\ CAD$ meeting $BC$ produced in $E$. If $AB\ =\ 10\ cm$, $AC\ =\ 6\ cm$, and $BC\ =\ 12\ cm$, find $CE$.

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Given:


In the given figure, $AE$ is the bisector of the exterior $∠\ CAD$ meeting $BC$ produced in $E$.


$AB\ =\ 10\ cm$, $AC\ =\ 6\ cm$, and $BC\ =\ 12\ cm$.


To do:


We have to find the measure of $CE$.


Solution:


$AE$ is the bisector of $∠\ CAD$, this implies,


$\angle BAD=\angle CAD$


We know that,


The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. 


Therefore,


$\frac{BE}{CE} = \frac{AB}{CA}$


$\frac{12+x}{x} = \frac{10}{6}$


$6(12+x) = 10(x)$


$72+6x = 10x$


$10x-6x=72$

$4x=72$

$x=\frac{72}{4}$

$x=18\ cm$


The measure of $CE$ is $18\ cm$.   

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Updated on: 10-Oct-2022

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