Question

A figure 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

Selected: ∠TRI, ∠PTR, and ∠OTR.

Reasons:
- TR is a radius to the point of tangency R, so it is perpendicular to the tangent (RX). Hence any angle at R formed by TR and the tangent (such as ∠TRI) is 90°.
- PO is a diameter through the center T, so TP and TO are collinear (opposite) directions. TR (to the top point R) is perpendicular to that diameter, so the central angles at T formed with TR and either TP or TO (∠PTR and ∠OTR) are right angles.