+58 Points T and U lie on a circle centered at O, and point P is outside the circle such that PT and PU are tangent to the circle. If angle TPO = 33 degrees, then what is the measure of minor arc TU, in degrees?
+1222 Finding the Measure of Minor Arc TU
Understanding the Problem
We have a circle with center O.
Points T and U lie on the circle.
Point P is outside the circle with PT and PU tangent to the circle.
Angle TPO = 33 degrees.
We need to find the measure of minor arc TU.
Solution
Key Concept: The angle formed by two tangents to a circle from an external point is equal to the difference between the intercepted arcs.
Steps:
Find angle TPU:
Since PT and PU are tangents to the circle, angle PTO and angle PUO are right angles (90 degrees each).
In triangle TPU, the sum of angles is 180 degrees.
So, angle TPU = 180 - angle PTO - angle PUO = 180 - 90 - 90 = 0 degrees.
This means that points P, T, and U are collinear.
Find angle TOP:
Since triangle OTP is a right triangle (OT is a radius perpendicular to tangent PT), angle TOP = 90 - angle TPO = 90 - 33 = 57 degrees.
Find the measure of minor arc TU:
Angle TOP is the central angle subtended by minor arc TU.
The measure of a central angle is equal to the measure of its intercepted arc.
Therefore, the measure of minor arc TU is 57 degrees.
So, the measure of minor arc TU is 57 degrees.
