Secants: Proof Hint
- Tim Brzezinski
In the applet below, two secant segments are drawn to a circle from a point outside the circle. Interact with the applet below for a few minutes, then answer the questions that follow.
Questions: 1) What can you conclude about the measure of the pink angles? 2) Why can you conclude this? (If you need a hint, see the worksheet found here: https://tube.geogebra.org/m/dzksdCfS) 3) Note that the blue angle is congruent to itself. What property justifies this? 4) Move the slider all the way to the end one more time. What can you conclude about triangle ABC and triangle DEC? What previously learned theorem justifies this fact? 5) Use your result from (4) to write a relationship (i.e. equation) among the lengths a, b, ext_a, and ext_b. 6) Rewrite the equation you wrote in (5) above so that each side of the equation is written as a product. 7) Now, reset the slider again. Then, drag point A upwards along the circle so that it lies right on top of point E. (This now causes the secant segment to turn into a tangent segment.) Reslide the slider again one more time. What can you conclude about the lengths a and ext_a in this special scenario? 8) Answer questions (4) - (5) again within this context.