Observe how the angles a radius makes with a line change as the line shifts from a secant to a tangent.
[i]Answer these open ended questions on your own or with others to form deeper math connections.[/i]
Why are the angles shown in the figure congruent?
As you move the two points closer together, what happens to the measures of the two angles formed by the radii and the line?
What do you observe about these angles at the exact moment the two points coincide and the line becomes a tangent?
If you pick a point on the tangent line (other than the point of tangency), would its distance to the center be greater than, less than, or equal to the radius? Why?
Use your answer to the latest question to formally prove that the tangent line is perpendicular to the radius.