Question

Given a line segment XZ of length 4. A point Y divides XZ such that

\[\frac{ XY }{ YZ }=\frac{ 4 }{ \sqrt{32} }\]

Find the lengths of XY and YZ.

Answer 1

First, reduce the ratio. Then add the segment ratios together.

Remember, XY + YZ = XZ

Answer 2

\[\frac{4}{\sqrt{32}} = \frac{4}{\sqrt{(16)(2)}} = \frac{4}{4\sqrt{2}}= \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]

Answer 3

So \[XY + YZ = \sqrt{2}x + 2x = 4\]

Answer 4

So \[x=-2(\sqrt{2}-2)\]
Then just plug that back in to the equation and I have my lengths.

Answer 5

\[XY=4\sqrt{2}-4\]
\[YZ=-4(\sqrt{2}-2)\]

Thanks so much!