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The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 14$ and $WX = 6,$ find $YZ.$ Dont have a pic. And it not 8/3,66/7, or 63/5. Please write answer as fraction.
 Apr 7, 2023
 #1
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First, note that since X lies on the circumcircle of triangle WXY, we have angle(WXY) = angle(XZY), where ZY is a tangent to the circumcircle at X. Also, since WZ is parallel to ZY, we have angle(WZX) = angle(XZY).

Using the fact that the sum of angles in a triangle is 180 degrees, we can write:

angle(WXY) + angle(WZX) + angle(XZY) = 180 degrees

Substituting angle(WXY) = angle(XZY), we get:

2 angle(XZY) + angle(WZX) = 180 degrees

Since angle(WZX) = angle(XZY), we have:

3 angle(XZY) = 180 degrees

So, angle(XZY) = 60 degrees.

Now, consider triangle WXY. Using the law of cosines, we have:

XY^2 = WX^2 + WY^2 - 2 WX WY cos(angle(WXY))

Substituting XY = 14, WX = 6, and angle(WXY) = 120 degrees (since angle(WXY) + angle(XWY) + angle(XYW) = 180 degrees), we get:

14^2 = 6^2 + WY^2 - 2 * 6 * WY * (-1/2)

Simplifying, we get:

WY^2 + 6 WY - 40 = 0

Factoring, we get:

(WY - 4)(WY + 10) = 0

Since WY cannot be negative, we have WY = 4.

Then from similar triangles XWZ and YWX, YZ = 4*6/14 = 12/7.

 Apr 7, 2023
 #2
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wrong
Guest Apr 7, 2023

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