A radius and a tangent to a circle

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Question from elizabeth, a student:

show that the radius of a circle meets a tangent line to the circle in a 90 degree angle.
hint: start by assuming they are not perpendicular and at a contradiction.

Hi Elizabeth,

Suppose that PQ is a diameter of a circle and construct the tangent line to the circle at P. Construct a line from Q to meet the tangent line at R.

circle

If the tangent line does not make a right angle with the diameter QP then P and R are different points. Since QP is a diagonal to the circle and angle PRQ is a right angle, what can you say about the point R?

Penny

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