Question

in the figure below, z is the center of the circle. suppose that ( qr = 4x - 2 ), ( st = 10 ), ( zu = 8 ), and ( zv = 8 ). find the following.
( x = )
( vs = )

Explanation:

Step1: Recognize equal chords

Since \( ZU = ZV = 8 \) and both are perpendicular to chords \( QR \) and \( ST \) respectively, chords \( QR \) and \( ST \) are equal in length (chords equidistant from the center are equal). So \( QR = ST \).
Given \( ST = 10 \) and \( QR = 4x - 2 \), we set up the equation:
\( 4x - 2 = 10 \)

Step2: Solve for \( x \)

Add 2 to both sides:
\( 4x = 10 + 2 = 12 \)
Divide both sides by 4:
\( x = \frac{12}{4} = 3 \)

Step3: Find \( VS \)

Since \( ZV \) is perpendicular to \( ST \), it bisects \( ST \) (a perpendicular from the center to a chord bisects the chord). So \( VS = \frac{ST}{2} \).
Given \( ST = 10 \), then \( VS = \frac{10}{2} = 5 \)

Answer:

\( x = 3 \)
\( VS = 5 \)