Question

if yu = yv, st = 16, arc qs = 34°, and arc rt = 98°, find each measure.
qu = type your answer...
qr = type your answer...
arc st = type your answer...
arc qr = type your answer...
arc xt = type your answer...

Explanation:

Step1: Use the property of chords equidistant from the center

Since \(YU = YV\), chords \(QS\) and \(ST\) are equidistant from the center \(Y\). So, \(QU=\frac{1}{2}QS\). Given \(QS = 34^{\circ}\), then \(QU=\frac{34}{2}=17\)

Step2: Use the property of congruent chords

Since \(YU = YV\), chords \(QS\) and \(ST\) are congruent. So, \(arc\ QS=arc\ ST = 34^{\circ}\)

Step3: Use the property of arcs in a circle

The measure of an arc formed by two adjacent arcs is the sum of their measures. Let's assume the circle has a total of \(360^{\circ}\). But for \(arc\ QR\), since \(YU = YV\) and using the symmetry of the circle (chords equidistant from the center have related arc - chord properties). However, if we consider the fact that if two chords are equidistant from the center, their subtended arcs have certain relationships. But more directly, if we assume the circle has arcs related to the perpendicular bisectors. Since \(YU = YV\), and if we consider the arcs related to the chords. But another approach: If \(arc\ ST=34^{\circ}\) and \(arc\ RT = 98^{\circ}\), and using the property that in a circle, if two chords are equidistant from the center, the arcs they subtend have some symmetry. But actually, since \(YU = YV\), chords \(QS\) and \(ST\) are congruent, so \(arc\ QS=arc\ ST = 34^{\circ}\). For \(arc\ QR\), if we assume the circle has arcs such that if we consider the perpendicular bisectors of the chords. But a better way: In a circle, if two chords are equidistant from the center, then the arcs they subtend are congruent. But actually, \(arc\ QR\) and \(arc\ ST\) have a relationship. Wait, no. If \(YU = YV\), and \(YQ = YS=YR = YT\) (radii). Since \(YU\perp QS\) and \(YV\perp ST\), \(QS = ST\) (chords equidistant from the center are congruent). So \(arc\ QS=arc\ ST = 34^{\circ}\). For \(arc\ QR\), if we consider the fact that \(arc\ RT = 98^{\circ}\), and the circle has \(360^{\circ}\). But actually, since \(YU = YV\), and using the property of perpendicular from the center to a chord bisects the chord. \(QS = ST\) (because \(YU = YV\) and \(YU\perp QS\), \(YV\perp ST\)). So \(arc\ QS=arc\ ST = 34^{\circ}\). For \(arc\ QR\), if we assume the circle has arcs such that \(arc\ QR=arc\ RT\) (by symmetry of the circle with respect to the line \(Y - X\) (the line passing through the center and perpendicular to the chords \(QS\) and \(ST\))). So \(arc\ QR = 98^{\circ}\). For \(arc\ XT\), since \(YV\perp ST\), \(arc\ XT=\frac{1}{2}arc\ ST\). So \(arc\ XT = 17^{\circ}\)

Answer:

\(QU = 8\) (Since \(ST = 16\) and \(QU=\frac{1}{2}QS\), and \(QS = ST\) (chords equidistant from the center are congruent)), \(QR = 16\) (chords equidistant from the center are congruent), \(arc\ ST=34^{\circ}\), \(arc\ QR = 98^{\circ}\), \(arc\ XT=17^{\circ}\)