Question

in the figure below, z is the center of the circle. suppose that qr = 12, st = 12, uz = 8, and vz = 2x - 6. find the following.

Explanation:

Step1: Recall circle - chord property

In a circle, if two chords are equal, the perpendicular distances from the center of the circle to the chords are equal. Since $QR = ST=12$, then $UZ = VZ$.

Step2: Set up the equation

Set $UZ = VZ$, so $8=2x - 6$.

Step3: Solve for $x$

Add 6 to both sides of the equation: $8 + 6=2x-6 + 6$, which simplifies to $14 = 2x$. Then divide both sides by 2: $x=\frac{14}{2}=7$.

Step4: Find $SV$

Since $Z$ is the center of the circle and $ZV$ is the perpendicular from the center to chord $ST$, $SV=\frac{ST}{2}$ (a perpendicular from the center of a circle to a chord bisects the chord). Given $ST = 12$, then $SV = 6$.

Answer:

$x = 7$
$SV = 6$