In a $Δ\ ABC$, $AD$ is the bisector of $∠\ A$, meeting side $BC$ at $D$.If $AB\ =\ 10\ cm$, $AC\ =\ 6\ cm$, and $BC\ =\ 12\ cm$, find $BD$ and $DC$.

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Given:

In a $Δ\ ABC$, $AD$ is the bisector of $∠\ A$, meeting side $BC$ at $D$.

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

To do:

We have to find the measure of $BD$ and $DC$.

Solution:

$AD$ is the bisector of $∠\ A$, this implies,

$\angle BAD=\angle CAD$

We know that,

The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle. 

Therefore,

$\frac{AB}{AC} = \frac{BD}{DC}$

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

$(12-x)10 = x\times 6$

$120-10x = 6x$

$10x+6x=120$

$x=\frac{120}{16}$

$x=7.5\ cm$

The measure of $BD$ is $7.5\ cm$. 

The measure of $DC$ is $(12-7.5)\ cm=4.5\ cm$.

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Updated on: 10-Oct-2022

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