Question

in the figure below, z is the center of the circle. suppose that qr = 2x - 3, st = 11, zu = 5, and zv = 5. find the following.

Explanation:

Step1: Use the property of chords equidistant from the center

In a circle, if two chords are equidistant from the center, then the chords are congruent. Since \(ZU = ZV=5\), then \(QR = ST\).
Given \(QR = 2x - 3\) and \(ST = 11\), we set up the equation \(2x-3=11\).

Step2: Solve the equation for \(x\)

Add \(3\) to both sides of the equation \(2x-3 = 11\):
\(2x-3+3=11 + 3\), which simplifies to \(2x=14\).
Divide both sides by \(2\): \(x=\frac{14}{2}=7\).

Step3: Use the property of perpendicular from the center to a chord

The perpendicular from the center of a circle to a chord bisects the chord. Since \(ZY\perp ST\), then \(VS=\frac{1}{2}ST\).
Given \(ST = 11\), so \(VS=\frac{11}{2}=5.5\).

Answer:

\(x = 7\)
\(VS=5.5\)