Thales’ Theorem

New York State Common Core Math Geometry, Module 5, Lesson 1

Worksheets for Geometry

Student Outcomes

• Using observations from a pushing puzzle, explore the converse of Thales’ theorem: If triangle ABC is a right triangle, then A, B, and C are three distinct points on a circle with segment AC a diameter.
• Prove the statement of Thales’ theorem: If A, B, and C are three different points on a circle with segment AC a diameter, then angle ABC is a right angle.

Thales’ Theorem

Classwork

Opening Exercise

a. Mark points 𝐴 and 𝐵 on the sheet of white paper provided by your teacher.
b. Take the colored paper provided, and push that paper up between points 𝐴 and 𝐵 on the white sheet.
c. Mark on the white paper the location of the corner of the colored paper, using a different color than black. Mark that point 𝐶. See the example below.
d. Do this again, pushing the corner of the colored paper up between the black points but at a different angle. Again, mark the location of the corner. Mark this point 𝐷.
e. Do this again and then again.

• Show Video

Exploratory Challenge

Choose one of the colored points (𝐶, 𝐷, …) that you marked. Draw the right triangle formed by the line segment connecting the original two points 𝐴 and 𝐵 and that colored point. Take a copy of the triangle, and rotate it 180° about the midpoint of 𝐴𝐵.
Label the acute angles in the original triangle as 𝑥 and 𝑦, and label the corresponding angles in the rotated triangle the same.
Todd says 𝐴𝐶𝐵𝐶′ is a rectangle. Maryam says 𝐴𝐶𝐵𝐶′ is a quadrilateral, but she is not sure it is a rectangle.
Todd is right but does not know how to explain himself to Maryam. Can you help him out?
a. What composite figure is formed by the two triangles? How would you prove it?
i. What is the sum of the measures of 𝑥 and 𝑦? Why?
ii. How do we know that the figure whose vertices are the colored points (𝐶, 𝐷, …) and points 𝐴 and 𝐵 is a rectangle?
b. Draw the two diagonals of the rectangle. Where is the midpoint of the segment connecting the two original points 𝐴 and 𝐵? Why?
c. Label the intersection of the diagonals as point 𝑃. How does the distance from point 𝑃 to a colored point (𝐶, 𝐷, …) compare to the distance from 𝑃 to points 𝐴 and 𝐵?
d. Choose another colored point, and construct a rectangle using the same process you followed before. Draw the two diagonals of the new rectangle. How do the diagonals of the new and old rectangle compare? How do you know?
e. How does your drawing demonstrate that all the colored points you marked do indeed lie on a circle?

Example

In the Exploratory Challenge, you proved the converse of a famous theorem in geometry. Thales’ theorem states the following: If 𝐴, 𝐵, and 𝐶 are three distinct points on a circle, and 𝐴𝐵 is a diameter of the circle, then ∠𝐴𝐶𝐵 is right.
Notice that, in the proof in the Exploratory Challenge, you started with a right angle (the corner of the colored paper)
and created a circle. With Thales’ theorem, you must start with the circle and then create a right angle. Prove Thales’ theorem.
a. Draw circle 𝑃 with distinct points 𝐴, 𝐵, and 𝐶 on the circle and diameter 𝐴𝐵. Prove that ∠𝐴𝐶𝐵 is a right angle.
b. Draw a third radius (𝑃𝐶). What types of triangles are △ 𝐴𝑃𝐶 and △ 𝐵𝑃𝐶? How do you know?
c. Using the diagram that you just created, develop a strategy to prove Thales’ theorem.
d. Label the base angles of △ 𝐴𝑃𝐶 as 𝑏° and the base angles of △ 𝐵𝑃𝐶 as 𝑎°. Express the measure of ∠𝐴𝐶𝐵 in terms of 𝑎° and 𝑏°.
e. How can the previous conclusion be used to prove that ∠𝐴𝐶𝐵 is a right angle?8

Exercises

1. 𝐴𝐵 is a diameter of the circle shown. The radius is 12.5 cm, and 𝐴𝐶 = 7 cm.
   a. Find 𝑚∠𝐶.
   b. Find 𝐴𝐵.
   c. Find 𝐵𝐶.
   
2. In the circle shown, 𝐵𝐶 is a diameter with center 𝐴.
   a. Find 𝑚∠𝐷𝐴𝐵.
   b. Find 𝑚∠𝐵𝐴𝐸.
   c. Find 𝑚∠𝐷𝐴𝐸.

Lesson Summary

Theorems:

THALES’ THEOREM: If 𝐴, 𝐵, and 𝐶 are three different points on a circle with a diameter ̅𝐴𝐵̅̅̅ , then ∠𝐴𝐶𝐵 is a right angle.

CONVERSE OF THALES’ THEOREM: If △ 𝐴𝐵𝐶 is a right triangle with ∠𝐶 the right angle, then 𝐴, 𝐵, and 𝐶 are three distinct points on a circle with a diameter 𝐴𝐵.
Therefore, given distinct points 𝐴, 𝐵, and 𝐶 on a circle, △ 𝐴𝐵𝐶 is a right triangle with ∠𝐶 the right angle if and only if 𝐴𝐵 is a diameter of the circle.

• Given two points 𝐴 and 𝐵, let point 𝑃 be the midpoint between them. If 𝐶 is a point such that ∠𝐴𝐶𝐵 is right, then 𝐵𝑃 = 𝐴𝑃 = 𝐶𝑃.

Relevant Vocabulary

CIRCLE: Given a point 𝐶 in the plane and a number 𝑟 > 0, the circle with center 𝐶 and radius 𝑟 is the set of all points in the plane that are distance 𝑟 from the point 𝐶.

RADIUS*: May refer either to the line segment joining the center of a circle with any point on that circle (a radius) or to the length of this line segment (the radius).

DIAMETER: May refer either to the segment that passes through the center of a circle whose endpoints lie on the circle (a diameter) or to the length of this line segment (the diameter).

CHORD: Given a circle 𝐶, and let 𝑃 and 𝑄 be points on 𝐶. 𝑃𝑄 is called a chord of 𝐶.

CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.

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