G-C Right triangles inscribed in circles I


Alignments to Content Standards: G-C.A.2

Task

Suppose $\overline{AB}$ is a diameter of a circle and $C$ is a point on the circle different from $A$ and $B$ as in the picture below:

Inscribegtriangle1_8dc943baed104b0e567afff7f5b362ef

  1. Show that triangles $COB$ and $COA$ are both isosceles triangles.
  2. Use part (a) and the fact that the sum of the angles in triangle $ABC$ is $180$ degrees to show that angle $C$ is a right angle.

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.

Solution

  1. Below is a picture with segment $\overline{OC}$ added:

    Inscribetriangle2_ada1146b3dc8f4a0bd959261da7bfdab

    Since segments $\overline{OB}$, $\overline{OC}$, and $\overline{OA}$ are all radii of the same circle, they are all congruent. Therefore both triangles $COB$ and $COA$ are isosceles triangles.

  2. 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 $$ m(\angle A) + m(\angle B) + m(\angle C) = 180. $$ Using the angle equivalences from the previous paragraph

    \begin{align} m(\angle C) &= m(\angle BCO) + m (\angle ACO) \\ & = m(\angle CBO) + m(\angle CAO) \\ & = m(\angle B) + m(\angle A) \end{align}
    Substituting this into the previous formula we find $$ 2m(\angle A) + 2m( \angle B) = 180 $$ and so $m(\angle A) + m(\angle B) = 90$. Since the measures of the three angles of triangle $ABC$ add to $180$ this means that $m(\angle C) = 90$ and so triangle $ABC$ is a right triangle as desired.

Amy Callahan says:

11 months

The first paragraph of the commentary states that triangles inscribed in CIRCLES are always right, this wording needs to be change to SEMI-CIRCLES.

Michael Nakamaye says:

11 months

yep! I changed the language-- thanks for catching this!

Daniel says:

about 1 year

In solution 1, step (b), "=m(∠B)+m(∠C)" I think you meant to put m(∠A) where you put m(∠C)

Michael Nakamaye says:

about 1 year

indeed I did-- thank you for catching this!!!

wright@math.byu.edu says:

over 2 years

removed

Michael Nakamaye says:

over 2 years

Thank you for your valuable suggestions and corrections (and also for the encouragement!). I will take care of the corrections and look into adding a second solution as soon as I have time.