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)

Question

Responses

62°
62°

87°
87°

25°

Answer 1

To solve the problem, we need to use the properties of inscribed angles and their corresponding arcs in a circle.

1. Understanding the Configuration:

   • Triangle XYZ is inscribed in a circle with diameter XZ. By the inscribed angle theorem, any angle inscribed in a circle that subtends a diameter is a right angle. Therefore, angle Y (the angle at vertex Y of triangle XYZ) is a right angle (90°).

2. Given Information:

   • Angle X = 59°.
   • Angle Y = 90° (since it's subtended by the diameter XZ).
   • We can find angle Z using the fact that the sum of the angles in a triangle is 180°: \[ Angle Z = 180° - Angle X - Angle Y = 180° - 59° - 90° = 31°. \]

3. Finding Arc XY:

   • The measure of an arc that subtends an inscribed angle is twice the inscribed angle. Since angle Y subtends arc XY, we have: \[ Arc XY = 2 \times Angle Y = 2 \times 90° = 180°. \]

4. Determining the Measure of Arc XY:

   • However, we're interested in the arc measure between points X and Y, which is subtended by angle Z at point Y.
   • The measure of arc XY corresponds to angle Z: \[ Arc XY = 2 \times Angle Z = 2 \times 31° = 62°. \]

Thus, the arc measure of XY is \( \boxed{62°} \).