Question

given the circle below with chords \\(\overline{vw}\\) and \\(\overline{xy}\\). find the length of \\(\overline{xz}\\). round to the nearest tenth if necessary.

Explanation:

Step1: Recall Chord Intersection Theorem

When two chords intersect in a circle, the products of the lengths of their segments are equal. So, \( VZ \times ZW = XZ \times ZY \).

Step2: Assign Known Values

Let \( XZ = x \), \( ZY = 34 - x \) (since \( XY = 34 \)), \( VZ = 32 \), \( ZW = 15 \).
Substitute into the theorem: \( 32 \times 15 = x \times (34 - x) \).

Step3: Solve the Equation

Simplify left side: \( 480 = 34x - x^2 \).
Rearrange: \( x^2 - 34x + 480 = 0 \).
Factor (or use quadratic formula). Factoring: \( (x - 16)(x - 30) = 0 \)? Wait, no, quadratic formula: \( x = \frac{34 \pm \sqrt{34^2 - 4 \times 1 \times 480}}{2 \times 1} \).
Calculate discriminant: \( 1156 - 1920 = -764 \)? Wait, mistake! Wait, \( ZY \) is not \( 34 - x \). Wait, the chord \( XY \) has length \( XZ + ZY = x + ZY = 34 \)? No, wait the diagram: \( XY \) is a chord with length from \( X \) to \( Y \), passing through \( Z \). So \( XZ + ZY = XY \), but \( VZ = 32 \), \( ZW = 15 \), so \( VZ \times ZW = XZ \times ZY \). Wait, \( VZ = 32 \), \( ZW = 15 \), so \( 32 \times 15 = XZ \times ZY \). And \( XY = XZ + ZY = 34 \). Let \( XZ = x \), then \( ZY = 34 - x \). So \( 32 \times 15 = x(34 - x) \). Wait, \( 32*15=480 \), so \( 480 = 34x - x^2 \), \( x^2 -34x +480=0 \). Discriminant: \( 34^2 - 4*1*480 = 1156 - 1920 = -764 \). That can't be. Wait, maybe I misassigned the segments. Wait, the chord \( VW \): \( VZ = 32 \), \( ZW = 15 \), so \( VW = 32 + 15 = 47 \)? No, no, the chords are \( VW \) and \( XY \), intersecting at \( Z \). So the theorem is \( VZ \times ZW = XZ \times ZY \). So \( VZ = 32 \), \( ZW = 15 \), so \( 32 \times 15 = XZ \times ZY \). And \( XY = XZ + ZY = 34 \)? Wait, no, the length of \( XY \) is \( XZ + ZY = 34 \)? Wait the diagram shows \( XY \) as a chord with length 34? Wait the number 34 is next to \( XY \)? Wait the diagram: \( VZ = 32 \), \( ZY = 34 - XZ \)? Wait no, maybe the 34 is the length of \( XY \)? Wait, no, the labels: \( V \) to \( Z \) is 32, \( Z \) to \( Y \) is... Wait, maybe I got the segments wrong. Let's re-express: Chords \( VW \) and \( XY \) intersect at \( Z \). So \( VZ \times ZW = XZ \times ZY \). Let \( XZ = x \), \( ZY = y \). Then \( 32 \times 15 = x \times y \), and \( x + y = 34 \) (since \( XY = x + y = 34 \)). So we have \( xy = 480 \) and \( x + y = 34 \). So solving \( x(34 - x) = 480 \) → \( 34x - x² = 480 \) → \( x² -34x +480 = 0 \). Wait, discriminant is negative, which is impossible. So I must have misread the diagram. Wait, maybe \( XY \) is not 34, but \( VY \) or something else. Wait, the diagram: \( V \) to \( Y \) is a chord? No, \( XY \) is a chord with length from \( X \) to \( Y \), passing through \( Z \), with \( ZY = 34 - XZ \)? Wait, no, maybe the 34 is the length of \( VZ + ZY \)? No, \( VZ \) is 32, \( ZY \) is 34? Wait, no, the numbers: 32 is \( VZ \), 34 is \( ZY \)? Wait, maybe the diagram has \( VZ = 32 \), \( ZY = 34 \), \( ZW = 15 \), and we need \( XZ \). Then the theorem is \( VZ \times ZW = XZ \times ZY \). So \( 32 \times 15 = XZ \times 34 \). Then \( XZ = \frac{32 \times 15}{34} = \frac{480}{34} ≈ 14.1 \)? Wait, that makes sense. Oh! I misread the diagram. The chord \( XY \): \( ZY = 34 \), not \( XY = 34 \). So \( VZ = 32 \), \( ZW = 15 \), \( ZY = 34 \), and we need \( XZ \). Then by chord intersection theorem: \( VZ \times ZW = XZ \times ZY \). So \( 32 \times 15 = XZ \times 34 \). Then \( XZ = \frac{32 \times 15}{34} = \frac{480}{34} ≈ 14.1 \). Yes, that works. So my initial mistake was misassigning \( ZY \)'s length. So correct step:

Step1: Apply Chord Intersection Theorem

When two chords intersect, \( VZ \times ZW = XZ \times ZY \).

Step2: Substitute Known Values

\( VZ = 32 \), \( ZW = 15 \), \( ZY = 34 \). So \( 32 \times 15 = XZ \times 34 \).

Step3: Solve for \( XZ \)

\( XZ = \frac{32 \times 15}{34} = \frac{480}{34} ≈ 14.1 \).

Answer:

\( \boxed{14.1} \)