+1439 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 $YZ$.
+195 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.
