SOLUTION: .Suppose that MP is a diameter of a circle centered at O, and Q is any other point on the circle. Draw the line through O that is parallel to MQ, and let R be the point where it me

Question 1155147: .Suppose that MP is a diameter of a circle centered at O, and Q is any other point on the circle. Draw the line through O that is parallel to MQ, and let R be the point where it meets minor arc PQ. Prove that R is the midpoint of minor arc PQ.

Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!

.
In triangle QOM QO = OM ( radii of the circle) Isosceles triangle
1=2
MQ || RM OQ is the transversal
therefore 2=3 (alternate angles)
So 1=3
MQ || RM OQ is the transversal
1=4 (corresponding angles)
Therefore 3=4
The central angles QOR and ROP are congruent.
Therefore measure of arc QR = measure arc PR
Hence R is the midpoint of minor arc PQ.

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