Prove the Pythagorean Theorem Using Similarity

New York State Common Core Math Geometry, Module 2, Lesson 24

Worksheets for Geometry

Student Outcomes

• Students prove the Pythagorean Theorem using similarity.
• Students use similarity and the Pythagorean Theorem to find the unknown side lengths of a right triangle.
• Students are familiar with the ratios of the sides of special right triangles with angle measures 45–45–90 and 30–60–90.

Prove the Pythagorean Theorem Using Similarity

Classwork

Exercises 1–3

Simplify as much as possible.

1. Find the length of the hypotenuse of a right triangle whose legs have lengths 50 and 100.
2. Can you think of a simpler method for finding the length of the hypotenuse in Exercise 1? Explain.
3. Find the length of the hypotenuse of a right triangle whose legs have lengths 75 and 225.

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Exploratory Challenge/Exercises 4–5

4. An equilateral triangle has sides of length 2 and angle measures of 60°, as shown below. The altitude from one vertex to the opposite side divides the triangle into two right triangles.
   a. Are those triangles congruent? Explain.
   b. What is the length of the shorter leg of each of the right triangles? Explain.
   c. Use the Pythagorean theorem to determine the length of the altitude.
   d. Write the ratio that represents shorter leg: hypotenuse.
   e. Write the ratio that represents longer leg: hypotenuse.
   f. Write the ratio that represents shorter leg: longer leg.
   g. By the AA criterion, any triangles with measures 30–60–90 will be similar to this triangle. If a 30–60–90 triangle has a hypotenuse of length 16, what are the lengths of the legs?
5. An isosceles right triangle has leg lengths of 1, as shown
   a. What are the measures of the other two angles? Explain.
   b. Use the Pythagorean theorem to determine the length of the hypotenuse of the right triangle.
   c. Is it necessary to write all three ratios: shorter leg: hypotenuse, longer leg: hypotenuse, and shorter leg: longer leg? Explain.
   d. Write the ratio that represents leg: hypotenuse.
   e. By the AA criterion, any triangles with measures 45–45–90 will be similar to this triangle. If a 45–45–90 triangle has a hypotenuse of length 20, what are the lengths of the legs?

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