Question

10 fill in the blank 1 point
if yu = yv, st = 16, arc qs = 34°, and arc rt = 98°, find each measure.
qu = 65
qr = 130
arc st = type your answer...
arc qr = type your answer...
arc xt = type your answer...

Explanation:

Step1: Use the property of perpendicular chords from the center

If a radius is perpendicular to a chord, it bisects the chord. Since \(YU = YV\), chords \(QR\) and \(ST\) are equidistant from the center \(Y\). So, the arcs they subtend have certain relationships.

Step2: Find arc \(ST\)

We know that if two chords are equidistant from the center of a circle, then their subtended arcs are congruent. But here, we can also use the fact that the measure of an arc is related to the central - angle.
Since \(YV\perp ST\), arc \(ST\) is twice the measure of the arc from the end of the perpendicular to the chord. But another way: we know that the sum of arcs in a circle is \(360^{\circ}\). However, for the measure of arc \(ST\), since \(YV\perp ST\), arc \(ST = 98^{\circ}\) (because chords equidistant from the center subtend congruent arcs. Chord \(QR\) and \(ST\) are equidistant from \(Y\) (since \(YU = YV\) and \(YU\perp QR\), \(YV\perp ST\)), and arc \(RT = 98^{\circ}\) is part of the relationship. But more accurately, if a radius is perpendicular to a chord, it bisects the arc subtended by the chord.
Since \(YV\perp ST\), arc \(ST=98^{\circ}\) (because chords equidistant from the center subtend congruent arcs. Given \(YU = YV\) (\(YU\perp QR\), \(YV\perp ST\)), and if we assume the circle has some symmetry based on the perpendicular - from - the - center property for chords.

Step3: Find arc \(QR\)

Since \(YU\perp QR\), and using the property that if a radius is perpendicular to a chord, it bisects the arc subtended by the chord. Also, since chords \(QR\) and \(ST\) are equidistant from the center \(Y\) (\(YU = YV\)), arc \(QR = 130^{\circ}\) (already given in the filled - in part, but if we calculate: assume the sum of arcs \(QS+QR + RT+...=360^{\circ}\), but more directly from the chord - distance property).

Step4: Find arc \(XT\)

The measure of arc \(XT\): Since \(YV\perp ST\), the central angle for arc \(ST\) is \(98^{\circ}\). The measure of an arc subtended by a chord and the perpendicular from the center. Also, if we consider the fact that the sum of arcs around a circle is \(360^{\circ}\). But another approach: the measure of an arc \(XT\): Since \(YV\perp ST\), and if we assume the circle has a central - angle measure. The measure of arc \(XT=\frac{1}{2}(180 - 98)=41^{\circ}\) (because \(YV\) is a radius perpendicular to \(ST\), and if we consider the right - triangle formed at \(V\) and the central - angle relationship. The central angle for arc \(ST\) is \(98^{\circ}\), and the radius \(YX\) (assuming \(X\) is a point such that \(YV\) is part of the radius). The measure of arc \(XT\) is half of the non - arc \(ST\) part of the semi - circle. The semi - circle is \(180^{\circ}\), so arc \(XT=\frac{180 - 98}{2}=41^{\circ}\)

Answer:

arc \(ST = 98^{\circ}\), arc \(QR = 130^{\circ}\), arc \(XT=41^{\circ}\)