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Illustrative Mathematics G-C Right triangles inscribed in circles I Alignments to Content Standards: G-C.A.2 Task ⎯⎯⎯⎯⎯⎯⎯ Suppose AB is a diameter of a circle and C is a point on the circle different from B as in the picture below: A and a. Show that triangles COB and COA are both isosceles triangles. b. Use part (a) and the fact that the sum of the angles in triangle show that angle C is a right angle. ABC is 180 degrees to 1 Illustrative Mathematics IM Commentary This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle with one side of the triangle a diameter: the fact that these triangles are always right triangles is often referred to as Thales' theorem. It does not have a lot of formal prerequisites, just the knowledge that the sum of the three angles in a triangle is 180 degrees. This task can be made substantially more challenging by omitting the auxiliary construction indicated in part (a), or just slightly more difficult by removing some of the explicit guidance from part (b). It can also be further facilitated, if needed, by suggesting for students to study the three angles in triangle ABC and their sum. As written, the task is suitable for assessment or instruction. If used for instruction, the teacher may wish to make the problem more open ended as indicated in the previous paragraph. Edit this solution Solution ⎯⎯⎯⎯⎯⎯⎯⎯ a. Below is a picture with segment OC added: ⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯ 2 Illustrative Mathematics ⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯ Since segments OB , OC , and OA are all radii of the same circle, they are all congruent. Therefore both triangles COB and COA are isosceles triangles. b. Angles B and BCO are congruent since they are opposite congruent sides of the isosceles triangle COB. Similarly angles A and ACO are congruent because they are opposite congruent sides of the triangle COA. The sum of the measures of the three angles in a triangle is 180 degrees and so m(∠A) + m(∠B) + m(∠C) = 180. Using the angle equivalences from the previous paragraph m(∠C) = m(∠BCO) + m(∠ACO) = m(∠CBO) + m(∠CAO) = m(∠B) + m(∠A) Substituting this into the previous formula we find 2m(∠A) + 2m(∠B) = 180 and so m(∠A) + m(∠B) = 90. Since the measures of the three angles of triangle ABC add to 180 this means that m(∠C) = 90 and so triangle ABC is a right triangle as desired. G-C Right triangles inscribed in circles I Typeset May 4, 2016 at 20:44:47. Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License . 3