We are given a circle with center W. Two chords, YZ and UX, intersect at point V. We are given the following information: $YZ = 17$, $UX = 11$, and $m\angle UX = 80.6^\circ$. We need to find the length of $UV$, the measure of arc $UY$, and the length of $VW$.

1. Problem Description

We are given a circle with center W. Two chords, YZ and UX, intersect at point V. We are given the following information: $YZ = 17$, $UX = 11$, and $m\angle UX = 80.6^\circ$. We need to find the length of $UV$, the measure of arc $UY$, and the length of $VW$.

2. Solution Steps

a) Find the length of $UV$.

Since the chords $UZ$ and $YZ$ intersect, and we know $YZ = 17$ and $UX = 11$, and also that they intersect at a right angle, we can use the property that if two chords intersect at a point, then the product of the segments of one chord is equal to the product of the segments of the other chord. Since $\angle UVY$ is a right angle (90 degrees), $UV^2 + VY^2 = UY^2$ and $VX^2 + VZ^2 = XZ^2$. However, this information is not helpful in solving the problem.

Since $UV$ and $VZ$ intersect at right angles within the circle W, we can apply the following property: If two chords are perpendicular, the sum of the measures of the arcs intercepted by the chords is 180 degrees. Let $m\angle UY = x$. Then, $m\angle XZ = 180 - x$.

Since $UV \perp YZ$, $UY^2 + YX^2 = UX^2$ and $UZ^2 + XZ^2 = 4r^2$.

Let $UV = a$, $VX = b$, $YV = c$, $VZ = d$. Then, $a+b = 11$ and $c+d = 17$.

We need to find $a$. Since we do not have sufficient information, we cannot solve this problem.

However, if we assume $V$ is the midpoint of $UX$, then $UV = VX = 11/2 = 5.5$. This is an assumption, so we should state it. I will solve it under that assumption.

If $V$ is the midpoint of UX then $UV = UX/2 = 11/2 = 5.5$

b) Find the measure of arc $UY$.

Since we know that the chords intersect at a right angle at point V, we know that $m\angle UVY = 90^{\circ}$.

$m\angle UYX = 1/2 * m\angle UX$, so $m\angle UYX = 1/2 (80.6) = 40.3^{\circ}$

Also $m \angle UY = 180^\circ - 90^\circ - 40.3^\circ = 49.7^\circ$.

c) Find the length of VW.

We do not have enough information to calculate the length of $VW$. Since $\angle UVY = 90^{\circ}$, $UY$ and $XZ$ are diameters. However we don't know the radius of the circle.

3. Final Answer

a) $UV = 5.5$ (assuming $V$ is the midpoint of $UX$)

b) $m\angle UY = 49.7^\circ$

c) Cannot be determined without more information.