Question

Triangle XYZ is inscribed inside a circle with diameter XZ and the inscribed angle at Y. The angle measure at X is 59°. Using what you know about inscribed angles, find the arc measure of XY. (1 point) Responses 31° 31° 298° 298° 62° 62° 118°

Answer 1

To solve this problem, we need to use some properties of inscribed angles and the corresponding arc measures in a circle.

1. Angle at X is given as \( \angle X = 59^\circ \).
2. The inscribed angle \(\angle Y\) subtends arc \(XZ\). According to the inscribed angle theorem, the measure of the inscribed angle is half the measure of the arc that it subtends.

Since \(XZ\) is the diameter of the circle, the arc \(XZ\) corresponds to an angle of \(180^\circ\) at the center of the circle (as it is the angle subtended by the diameter).

Thus,
\[
\angle Y = \frac{1}{2} \text{(measure of arc } XZ) = \frac{1}{2} \times 180^\circ = 90^\circ.
\]

3. Since angle \(X\) is subtended from the arc \(YZ\), and we already know \(YZ\) must subtend an angle at the center, we find this angle at the center:
\[
\text{Arc } YZ = 2 \times \angle X = 2 \times 59^\circ = 118^\circ.
\]

4. Now, since the total arc measure around the circle is \(360^\circ\), we can find the measure of arc \(XY\) using the relationship:
\[
\text{Arc } XY + \text{Arc } YZ + \text{Arc } XZ = 360^\circ.
\]
Since arc \(XZ\) is \(180^\circ\):
\[
\text{Arc } XY + 118^\circ + 180^\circ = 360^\circ.
\]
5. Simplifying this gives us:
\[
\text{Arc } XY + 298^\circ = 360^\circ,
\]
\[
\text{Arc } XY = 360^\circ - 298^\circ = 62^\circ.
\]

Therefore, the arc measure of \(XY\) is \(62^\circ\).

So, the final answer is:
**62°**.