Question

10 fill in the blank 1 point
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 and perpendicular from the center

Since \(YU = YV\), the chords \(QR\) and \(ST\) are equidistant from the center \(Y\). So, \(QR=ST\). Given \(ST = 16\), then \(QU=\frac{QR}{2}\) (perpendicular from the center to a chord bisects the chord). So \(QU=\frac{16}{2}=8\).

Step2: Find the measure of arc \(ST\)

The measure of an arc is related to the central - angle subtended by it. Since \(YU = YV\), chords \(QR\) and \(ST\) are congruent. Congruent chords subtend congruent arcs. But we need to check the given arcs. Wait, another property: if a line from the center is perpendicular to a chord, it bisects the chord. Also, for the arcs, if two chords are equidistant from the center, their arcs are congruent. But we know that the measure of an arc is twice the inscribed - angle subtended by it (not directly here, but for congruent chords). Since \(QR\) and \(ST\) are congruent chords (because they are equidistant from the center \(Y\) as \(YU = YV\)), \(QR = ST=16\). The measure of arc \(ST\): we know that the sum of arcs in a circle is \(360^{\circ}\). But also, for a chord, if we consider the fact that the length of a chord \(l = 2r\sin\frac{\theta}{2}\) (where \(\theta\) is the central - angle subtended by the chord). But since \(QR\) and \(ST\) are congruent chords (equidistant from the center), and we are not using the chord - length formula in the most basic sense. Wait, no, another approach:
The measure of arc \(ST\): we know that for a circle, if two chords are equidistant from the center, their arcs are congruent. But we also know that the measure of an arc \(m\widehat{RT}=98^{\circ}\) and \(m\widehat{QS} = 34^{\circ}\). But for the chord \(ST\), since \(YV\perp ST\), the arc \(ST\) is related to the central - angle. Wait, no, the key property is: If a line from the center of a circle is perpendicular to a chord, then it bisects the chord. Also, if two chords are equidistant from the center of a circle, then the chords are congruent. So \(QR = ST\). And the measure of arc \(ST\): we know that if two chords are congruent, then their arcs are congruent (in the sense of length - related arc measure, but in terms of degree measure, if the circle is uniform). Wait, no, the degree measure of an arc is equal to the measure of the central - angle subtended by it. For congruent chords (equal length), the central - angles are equal. But we are given \(ST = 16\) (chord length). Since \(QR\) and \(ST\) are congruent chords (because \(YU = YV\)), \(QR = ST = 16\), \(QU=\frac{QR}{2}=8\).
For the arc measures:
- The measure of arc \(ST\): Since \(YV\perp ST\), and if we assume the circle has some symmetry. Wait, no, another property: the measure of an arc \(m\widehat{ST}\): we know that the sum of arcs \(m\widehat{QS}+m\widehat{QR}+m\widehat{RT}+m\widehat{ST}=360^{\circ}\). But we need to use the fact that \(QR\) and \(ST\) are congruent chords (so \(m\widehat{QR}=m\widehat{ST}\)). Let \(x=m\widehat{QR}=m\widehat{ST}\). Then \(34 + x+98 + x=360\).
\[2x+132 = 360\]
\[2x=360 - 132\]
\[2x = 228\]
\[x = 114^{\circ}\] for \(m\widehat{QR}\) and \(m\widehat{ST}\).
- For arc \(XT\): The measure of arc \(RT = 98^{\circ}\), and if we consider the fact that \(YV\perp ST\), the central - angle for arc \(ST\) is \(114^{\circ}\), so the central - angle for arc \(XT\): Since \(YV\perp ST\), the central - angle for arc \(ST\) is \(114^{\circ}\), and the central - angle for arc \(XT\) is \(\frac{1}{2}(m\widehat{ST})\) (because \(YV\) is perpendicular to \(ST\) and bisects the arc \(ST\) (a line from the center perpendicular to a chord bisects the arc subtended by the chord). So \(m\widehat{XT}=\frac{1}{2}m\widehat{ST}\). Since \(m\widehat{ST}=114^{\circ}\), \(m\widehat{XT}=57^{\circ}\).

Answer:

\(QU = 8\)
\(QR = 16\)
\(arc\ ST=114^{\circ}\)
\(arc\ QR = 114^{\circ}\)
\(arc\ XT=57^{\circ}\)