Question

in the figure below, z is the center of the circle. suppose that qr = 5x - 1, sr = 24, zu = 10, and zv = 10. find the following.

x =
vs =

Explanation:

Step1: Identify congruent chords

Since \( ZU = ZV = 10 \) (perpendicular distances from center \( Z \) to chords \( QR \) and \( SR \)), chords \( QR \) and \( SR \) are congruent.

Step2: Set up equation for \( x \)

Set \( QR = SR \):
\( 5x - 1 = 24 \)

Step3: Solve for \( x \)

Add 1 to both sides: \( 5x = 25 \)
Divide by 5: \( x = \frac{25}{5} = 5 \)

Step4: Find radius of the circle

First, find \( QU \): since \( ZU \perp QR \), \( QU = \frac{1}{2}QR \).
\( QR = 5(5)-1 = 24 \), so \( QU = 12 \).
Use Pythagoras in \( \triangle ZUQ \):
\( ZQ^2 = ZU^2 + QU^2 = 10^2 + 12^2 = 100 + 144 = 244 \), so \( ZQ = \sqrt{244} = 2\sqrt{61} \) (this is the radius, \( ZS = 2\sqrt{61} \))

Step5: Find \( VS \)

Since \( ZV \perp SR \), \( SV = \frac{1}{2}SR = 12 \) (we can also use Pythagoras in \( \triangle ZVS \):
\( VS^2 = ZS^2 - ZV^2 = 244 - 100 = 144 \)
Take square root: \( VS = \sqrt{144} = 12 \)

Answer:

\( x = 5 \)
\( VS = 12 \)