#2**+2 **

I have drawn this in GeoGebra and got the geometric solution of approx 9.4 (2 decimal places)

This does agree with guests answer.

Here is my pic.

You can click on the grab point G to change the size of the circle an you will see that the answer does not change (while the circle actually exists)

Melody Aug 2, 2020

#3**+1 **

Put another point V on the tangent, below X.

The angle VXY = angle XWY (angle on the circumference in the opposite segment), and also angle VXY = angle XZW (since the tangent at X is parallel to WZ), so angle XWY = angle XZW.

We have then two similar triangles, XWY, and XZW, (the angle at X is common to both triangles).

So,

\(\displaystyle \frac{WX}{XZ}=\frac{XY}{WX},\\ \displaystyle \frac{8}{XZ}=\frac{14}{8}.\)

\(\displaystyle XZ= \frac{64}{14}=\frac{32}{7},\\ \displaystyle YZ = 14 - \frac{32}{7}=\frac{98}{7}-\frac{32}{7}=\frac{66}{7} \approx9.42857.\)

Tiggsy Aug 2, 2020