Triangle inscribe in a semicircle.

[westin]

Triangle XYZ is inscribed in circle O as shown. If circle O has a diameter of 2 units and XZ = 2 units, what is the degree measure of XYZ? (attached is the question screenshot)

answer is 45.

I know that point Z is right in the middle of the circle as XZ is root 2.. that means OZ = OZ = 1. besides that, I don't know how to get angle XYZ? it seems that it can be any angles based on the limited information. However, i think i am mostly wrong but I can't think of other clues....

Thanks

 

[Dr.Peterson]

Triangle XYZ is inscribed in circle O as shown. If circle O has a diameter of 2 units and XZ = 2 units, what is the degree measure of XYZ? (attached is the question screenshot)

answer is 45.

I know that point Z is right in the middle of the circle as XZ is root 2.. that means OZ = OZ = 1. besides that, I don't know how to get angle XYZ? it seems that it can be any angles based on the limited information. However, i think i am mostly wrong but I can't think of other clues....

Thanks

What is the central angle of arc XZ?

What do you know about inscribed angles?

 

[westin]

having a new point called A on the other side of the diameter. then angle XAZ = 45 degrees which share the same arc XZ with angle xyz. meaning XYZ is also 45 degress. thanks!!!

 

[Dr.Peterson]

having a new point called A on the other side of the diameter. then angle XAZ = 45 degrees which share the same arc XZ with angle xyz. meaning XYZ is also 45 degrees. thanks!!!

Apparently you are aware that inscribed angles are all congruent, but not that they are half the central angle. See

Circle Theorems

Your description of where Z is ("right in the middle of the circle") implied that angle XOZ, the central angle for arc XZ, is 90 degrees, so any inscribed angle subtended by XZ is 90/2 = 45 degrees. Your alternate approach is valid, just a little slower.

 

[westin]

got it. thank you.