Circles and related segments and angles

[Cake]

Hey Guys,
I'm having some trouble with this problem.

so far I know that XY = 10.
I looked at the whole triangle first. Triangle XYZ.
I know that XZ is 6 so I applied the pythagoreans theorem which got me YZ = 8.
From this step I don't know what to do. I've tried looking at it different ways but nothing pops up

 

[stapel]

Can you do anything with the fact that Triangle WXY is isosceles?

 

[Cake]

Can you do anything with the fact that Triangle WXY is isosceles?

I was thinking of drawing an altitude from O to W. I drew it but I couldn't get anything out of it

 

[Cake]

Correct so far.

Hint: let w = WZ ; then WX = 8 - w

okay! SO, YZ = 8 and YW = 8 - w and since YW is congruent to XW that means that it is also 8 - w.

Now, should I apply the pythagoreans theorem?
(8-w)^2 + w^2 = 6^2
64 - 16w + w^2 + w^2 = 36
-16w + 2w^2 = -28
2w^2 - 16w + 28 = 0
4 + SQRT2.

Hmm, this shouldn't be it because the answer is 1.75
I'm close though. This looks like it was going the right way.

 

[Cake]

C'mon Cake...careful: hypotenuse is not XZ :shock:

My apologies.
w^2 + 6^2 = (8-w)^2
w^2 + 36 = 64 - 16w +w^2
36 = 64 - 16w
-28 = -16w
1.75 = w


\^_^/
thank you guys!