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The figure shows △ABC . BD is the angle bisector of ∠ABC .

What is AD ?

Enter your answer in the box as a fraction.

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The figure shows ABC BD is the angle bisector of ABC What is AD Enter your answer in the box as a fraction units class=

Respuesta :

WojtekR
From angle bisector theorem, we know that:

[tex]\dfrac{|AD|}{|DC|}=\dfrac{|BA|}{|BC|}\\\\\\\dfrac{|AD|}{|DC|}=\dfrac{8}{10}=\dfrac{4}{5}[/tex]

Moreover:

[tex]|AD|+|DC|=6[/tex]

so:

[tex]|DC|=6-|AD|[/tex]

and

[tex]\dfrac{|AD|}{|DC|}=\dfrac{4}{5}\\\\\\\dfrac{|AD|}{6-|AD|}=\dfrac{4}{5}\qquad\qquad\text{cross multiplying}\\\\\\ 5\cdot|AD|=4\cdot\big(6-|AD|\big)\\\\5\cdot|AD|=24-4\cdot|AD|\\\\5\cdot|AD|+4\cdot|AD|=24\\\\9\cdot|AD|=24\qquad|:9\\\\\\|AD|=\dfrac{24}{9}=\boxed{\dfrac{8}{3}}[/tex]

The length of AD is [tex]\dfrac{8}{3}[/tex].

Given:

BD is the angle bisector of the triangle ABC.

The figure of the triangle ABC.

To find:

The length of AD.

Explanation:

According to the Angle Bisector Theorem, the angle bisector of a triangle divides the opposite side in the same proportion of the other two sides.

Let [tex]x[/tex] be the length of AD. Then, the length of DC is [tex]6-x[/tex].

Using the Angle Bisector Theorem, we get

[tex]\dfrac{8}{10}=\dfrac{x}{6-x}[/tex]

[tex]\dfrac{4}{5}=\dfrac{x}{6-x}[/tex]

[tex]4(6-x)=5(x)[/tex]

[tex]24-4x=5x[/tex]

[tex]24=5x+4x[/tex]

[tex]24=9x[/tex]

[tex]\dfrac{24}{9}=x[/tex]

[tex]\dfrac{8}{3}=x[/tex]

Therefore, the length of AD is [tex]\dfrac{8}{3}[/tex].

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