Question

The figure shows △XYZ. XW⎯⎯⎯⎯⎯⎯ is the angle bisector of ∠YXZ .

What is WZ ?

Answer 1

3.84

Explanation:

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Answer 2

Use the law of cosines to find the measure of angle YXZ.


`8^2=6.5^2+6^2-2(6.5)(6)\cos(\angle YXZ)`

`\cos(\angle YXZ)\approx0.1827`

`\angle YXZ\approx 1.3871\text{ rad}\approx79.4734^\circ`

This means angles YXW and WXZ share the same measure of about
`39.7367^\circ`.

Use the law of cosines again to find the measure of angle XZW.


`6.5^2=8^2+6^2-2(8)(6)\cos(\angle XZW)`

`\cos(\angle XZW)\approx0.6016`

`\angle XZW\approx0.9253\text{ rad}\approx53.0181^\circ`

This means the measure of angle XWZ is


`180^\circ=\angle WXZ+\angle XWZ+\angle XZW\implies \angle XWZ\approx87.25^\circ`

Now using the law of sines, you have


`(\sin(\angle WXZ))/(WZ)=\frac{\sin(\angle XWZ)}6\implies WZ\approx3.84`