Question
Triangle XYZ is equilateral with vertices located on circle W.
Circle W is shown. Line segments W Y, W Z, and W X are radii. Lines are drawn to connect the points on the circle to form a triangle. Sides Z X, X Y, and Z Y are congruent.
Which measurements are correct? Select two options.
mArc X Y = 60°
mArc Y Z = 120°
mArc Z X = 180°
m∠XWZ = 60°
m∠YWZ = 120°
Circle W is shown. Line segments W Y, W Z, and W X are radii. Lines are drawn to connect the points on the circle to form a triangle. Sides Z X, X Y, and Z Y are congruent.
Which measurements are correct? Select two options.
mArc X Y = 60°
mArc Y Z = 120°
mArc Z X = 180°
m∠XWZ = 60°
m∠YWZ = 120°
Answers
GPT-4o mini
To analyze the measurements involving triangle XYZ, which is equilateral, we need to consider the properties of both an equilateral triangle and the associated arcs on circle W.
1. An equilateral triangle has all its internal angles equal to 60°. Thus, the angles at the center of the circle corresponding to each side of the triangle is also 60° when the angles are subtended by the sides of triangle XYZ.
2. The arc measures corresponding to these angles can be determined because arcs subtended by central angles are equal to the angle measure itself.
Now, let’s evaluate the options:
- **mArc XY = 60°**: This is true because the angle ∠XWZ is at the center of the circle and subtended by side XY, which is one side of the equilateral triangle.
- **mArc YZ = 120°**: If we look at the total circle (360°) and the other two arcs created by the triangle, we find that the combined angles mArc XY + mArc YZ + mArc ZX = 360°. Thus, mArc YZ would assume a value of 120°.
- **mArc ZX = 180°**: This cannot be true as we only have three arcs each taking a portion of the 360°.
- **m∠XWZ = 60°**: This is also true, being a central angle corresponding to side XY.
- **m∠YWZ = 120°**: This would be incorrect. Given that ∠XWZ is already accounted for, ∠YWZ would only add up to the total of 360° minus the previous angles.
Therefore, the two correct measurements are:
1. **mArc XY = 60°**
2. **m∠XWZ = 60°**
So the selected correct options should be:
1. mArc XY = 60°
2. m∠XWZ = 60°
1. An equilateral triangle has all its internal angles equal to 60°. Thus, the angles at the center of the circle corresponding to each side of the triangle is also 60° when the angles are subtended by the sides of triangle XYZ.
2. The arc measures corresponding to these angles can be determined because arcs subtended by central angles are equal to the angle measure itself.
Now, let’s evaluate the options:
- **mArc XY = 60°**: This is true because the angle ∠XWZ is at the center of the circle and subtended by side XY, which is one side of the equilateral triangle.
- **mArc YZ = 120°**: If we look at the total circle (360°) and the other two arcs created by the triangle, we find that the combined angles mArc XY + mArc YZ + mArc ZX = 360°. Thus, mArc YZ would assume a value of 120°.
- **mArc ZX = 180°**: This cannot be true as we only have three arcs each taking a portion of the 360°.
- **m∠XWZ = 60°**: This is also true, being a central angle corresponding to side XY.
- **m∠YWZ = 120°**: This would be incorrect. Given that ∠XWZ is already accounted for, ∠YWZ would only add up to the total of 360° minus the previous angles.
Therefore, the two correct measurements are:
1. **mArc XY = 60°**
2. **m∠XWZ = 60°**
So the selected correct options should be:
1. mArc XY = 60°
2. m∠XWZ = 60°