Topics
7. Angles & Triangles
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United States of America
Grade 8

Investigation: Discover the Pythagorean theorem

Lesson
Practice
Overview
Activity 1
Activity 2
Reflection

Investigate the Pythagorean Theorem and why it works.

Objectives
  • To prove an important relationship, namely the Pythagorean theorem, between the lengths of the sides of right triangles exists.
Materials
  • Plain paper
  • Pen or pencil
  • Colored pencils or markers
  • Straight edge
  • Scissors

Prove the Pythagorean Theorem - hands on activity

Key Vocab
Before you begin this activity, define each of these terms.
  • Right triangle
  • Hypotenuse
  • Leg
  • Area
  • Area of a Square
Why prove it?
The study of mathematics is like a huge pyramid, with things which are proven forming the base of more complicated work above it. Complete the following procedure to prove the Pythagorean Theorem.

Start with a square with four equal right triangles drawn into it, so that the hypotenuses form a square in the middle. Each of these triangles has sides a and b, and hypotenuse c.

Use colored pencils to make each of the four triangles a different color.

To be sure that the four triangles we are creating are equal, it may be helpful to trace the first one and create a stencil of it. Then you can use the stencil to form the other three triangles.

A square with four equal right triangles drawn into it, so that the hypotenuses form a square in the middle. Each of these triangles has sides a and b, and hypotenuse c.

Find the area of the white square in the center with its side length equal to c.

A square with four equal right triangles drawn into it, so that the hypotenuses form a square in the middle. Each of these triangles has sides a and b, and hypotenuse c. There is a label Area in the center of the square.

Cut out the four colored triangles and rearrange the triangles as shown in the diagram.

Then cut the white square that was in the middle of the first diagram so that it fits into the spaces available.

A square divided into two squares and two rectangles with diagonal markings.The first square has a side length of a and the second square has a side length of b. The two rectangles have lengths b and widths a.

Find the area of the two white squares with side lengths equal to a and b.

A square divided into two squares and two rectangles with diagonal markings.The first square has a side length of a and the second square has a side length of b. The two rectangles have lengths b and widths a. There are labels Area for each square
Investigate
Consider the following questions once you have completed the above procedure.
1.
Discuss how you calculated the area of the white square in the center with its side length equal to c with a partner.
2.
Discuss how you calculated the area of the two white squares with side lengths equal to a and b with a partner.
3.
Based upon your answers to the previous questions, as well as the procedure above, do you think that the equation a^{2} + b^{2} = c^{2} will always be true?

Prove the Pythagorean Theorem - applet activity

Another way to verify the Pythagorean Theorem, a^{2} + b^{2} = c^{2}, is true for any right triangle is by manipulating the applet below.

Manipulate the following applet to make the Pythagorean theorem make sense graphically.

Loading interactive...
Investigate
Consider the following questions once you have manipulated the applet.
1.
Write in words what you think is happening from the display the Geogebra applet gives. Then, discuss your answer with a partner.

Answer the questions below after you have completed the activity.

Discussion
1.
Which of the two activities showed the Pythagorean Theorem is always true the most clearly for you? Discuss with a partner why you felt the representation in the activity you selected made the most sense to you.

Outcomes

8.G.B.6

Explain a proof of the Pythagorean theorem and its converse.

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