Question

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)
º

Answer 1

In a circle, an inscribed angle is half the measure of the arc that it subtends. In triangle \( XYZ \), angle \( X \) is given as \( 59^\circ \).

The inscribed angle \( X \) subtends arc \( YZ \). Therefore, the arc measure of \( YZ \) is twice the measure of angle \( X \):

\[
\text{Arc } YZ = 2 \times \text{Angle } X = 2 \times 59^\circ = 118^\circ
\]

The entire circle is \( 360^\circ \). Since \( XZ \) is the diameter of the circle, the arc \( XY \) and the arc \( YZ \) together make up half of the circle (the semicircle):

\[
\text{Arc } XY + \text{Arc } YZ = 180^\circ
\]

Now, we can find the measure of arc \( XY \):

\[
\text{Arc } XY = 180^\circ - \text{Arc } YZ = 180^\circ - 118^\circ = 62^\circ
\]

Thus, the arc measure of \( XY \) is

\[
\boxed{62^\circ}
\]