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Let $X$, $Y$, and $Z$ be points on a circle. Let line $XY$ and the tangent to the circle at $Z$ intersect at $W$. If $\overline{WY} \perp \overline{WZ}$, then find $YZ$.

 Mar 20, 2024
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Analyze the Setup:

 

We have a circle with points X, Y, and Z on it.

 

Line XY intersects a tangent line drawn at point Z at point W.

 

We are given that WY=1 and YZ=1 (both lengths are the same).

 

We need to find the length XZ.

 

Utilize Properties of Tangents and Perpendicular Lines:

 

Since the tangent line is drawn at point Z, it's perpendicular to the radius drawn from the center of the circle (O) to point Z (radius OZ).

 

We are also given that WY⊥WZ. This implies that ∠WZY is a right angle.

 

Apply the Pythagorean Theorem:

 

Since we have a right angle at point Z (proven in step 2), triangle WYZ is a right triangle. We are given that the lengths of two legs, WY=1 and YZ=1.

 

 

By the Pythagorean Theorem:

 

WZ2=WY2+YZ2

 

WZ2=12+12

 

WZ2=2

 

Taking the square root of both sides (positive for lengths), we find:

 

WZ=2​

 

Recognize Symmetry and Solve for XZ:

 

Since the tangent line is perpendicular to the radius at the point of tangency (point Z), radius OZ bisects XW (symmetry). This means WX=WZ=2​.

 

Now, consider triangle XYZ. We know the lengths of two sides, YZ=1 and WZ=2​. Since these two segments form a right angle (proven in step 2), triangle XYZ is also a right triangle.

 

We can again apply the Pythagorean Theorem:

 

XZ2=YZ2+WZ2

 

XZ2=12+(2​)2

 

XZ2=1+2

 

XZ2=3

 

Taking the square root of both sides (positive for lengths), we find:

 

XZ=3​

 

Therefore, the length YZ is 3​​.

 Mar 20, 2024

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