Question

The figure below shows a circle with center TT, diameter start overline, P, O, end overline
PO
, and tangent R, X, with, left-right arrow, on top
RX
. Which of the angles must be right angles? Select all that apply.
T
R
P
O
Y
S
I
X
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Answer
angle, O, P, S∠OPS
angle, T, R, I∠TRI
angle, P, T, R∠PTR
angle, P, S, Y∠PSY
angle, O, T, R∠OTR
angle, P, Y, S∠PYS

Answer 1

I can’t confidently pick the correct choices without seeing exactly where S, Y and I are on the diagram. Can you either attach the figure (or a clearer picture) or tell me exactly which points lie on the circle, which point is the point of tangency, and which points lie on the tangent line?

Helpful facts you can use to identify the right angles (I’ll use the labels you gave):

- Radius-to-tangent is a right angle. If R is the point of tangency and I (or X) is any point on the tangent line, then the angle between TR and that tangent line is 90°. So any listed angle with vertex R that is formed by RT and a segment along the tangent (for example ∠TRI or ∠TRX) must be right.

- Thales / semicircle theorem: any angle with vertex at a point on the circle that subtends the diameter PO is a right angle. In other words, for any point Z on the circle (Z ≠ P,O), ∠PZO = 90°. So any listed angle whose vertex is a point on the circle and whose sides go to P and O (i.e. an angle of the form ∠PZO) must be right.

If you send the figure or tell me which listed angle(s) are of the two types above I’ll mark exactly which options to choose.