Lesson 6A Proof of the Pythagorean Theorem

Learning Goal

Let’s prove the Pythagorean Theorem.

Learning Targets

  • I can explain why the Pythagorean Theorem is true.

  • If I know the lengths of two sides, I can find the length of the third side in a right triangle.

  • When I have a right triangle, I can identify which side is the hypotenuse and which sides are the legs.

Lesson Terms

  • hypotenuse
  • legs
  • Pythagorean Theorem

Warm Up: Notice and Wonder: A Square and Four Triangles

Problem 1

A blue square with a white square inscribed with its vertices intersecting with the outside square's side midpoints. A 2nd white square with green triangles drawn from the sides.

What do you notice? What do you wonder?

Activity 1: Adding Up Areas

Problem 1

Both figures shown here are squares with a side length of . Notice that the first figure is divided into two squares and two rectangles. The second figure is divided into a square and four right triangles with legs of lengths and . Let’s call the hypotenuse of these triangles .

Two squares of the same area are labeled “F” and “G”. Square F is divided into the following: A square with side lengths “a”. Two rectangles with side lengths “a” and “b”. A square with side lengths “b”. Square G is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths.

  1. What is the total area of each figure?

  2. Find the area of each of the 9 smaller regions shown the figures and label them.

  3. Add up the area of the four regions in Figure F and set this expression equal to the sum of the areas of the five regions in Figure G. If you rewrite this equation using as few terms as possible, what do you have?

Are you ready for more?

Problem 1

Take a 3-4-5 right triangle, add on the squares of the side lengths, and form a hexagon by connecting vertices of the squares as in the image. What is the area of this hexagon?

A hexagon formed by a 3-4-5 right triangle, with the squares of the side lengths added on and connecting the vertices of the squares

Activity 2: Find the Missing Side Lengths

Problem 1

  1. Find .

    A right triangle with legs, a = square root of 10, b = square root of 40, and hypotenuse, c.

  2. Find .

    A right triangle with legs: square root of 8, b, and hypotenuse c = square root of 26,

  3. A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?

  4. A right triangle has a side of length and a hypotenuse of length . What is the length of the other side?

  5. Find the value of in the figure.

    Two right triangles sharing a leg. One triangle has leg 5 and hypotenuse square root of 23. Other has leg x and hypotenuse square root of 18.

Are you ready for more?

Problem 1

The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.

A spiral made by starting with a right triangle with both legs measuring one unit each and continuing to add triangles where the leg of one is the hypotenuse of the next.

Find the length, , of the hypotenuse of the last triangle constructed in the figure.

Activity 3: A Transformational Proof

Problem 1

Use the applets to explore the relationship between areas.

  • Consider Squares and .

  • Check the box to see the area divided into five pieces with a pair of segments.

  • Check the box to see the pieces.

  • Arrange the five pieces to fit inside Square .

  • Check the box to see the right triangle.

  1. Arrange the figures so the squares are adjacent to the sides of the triangle.

  2. If the right triangle has legs and and hypotenuse , what have you just demonstrated to be true?

    open applet in presentation mode

  3. Try it again with different squares. Estimate the areas of the new Squares, , , and and explain what you observe.

    open applet in presentation mode

  4. Estimate the areas of these new Squares, , , and , and then explain what you observe as you complete the activity.

    open applet in presentation mode

  5. What do you think we may be able to conclude?

    open applet in presentation mode
Print Version

Your teacher will give your group a sheet with 4 figures and a set of 5 cut out shapes labeled D, E, F, G, and H.

  1. Arrange the 5 cut out shapes to fit inside Figure 1. Check to see that the pieces also fit in the two smaller squares in Figure 4.

  2. Explain how you can transform the pieces arranged in Figure 1 to make an exact copy of Figure 2.

  3. Explain how you can transform the pieces arranged in Figure 2 to make an exact copy of Figure 3.

  4. Check to see that Figure 3 is congruent to the large square in Figure 4.

  5. If the right triangle in Figure 4 has legs and and hypotenuse , what have you just demonstrated to be true?

Lesson Summary

The figures shown here can be used to see why the Pythagorean Theorem is true. Both large squares have the same area, but they are broken up in different ways. (Can you see where the triangles in Square are located in Square ? What does that mean about the smaller squares in and ?) When the sum of the four areas in Square are set equal to the sum of the 5 areas in Square , the result is , where is the hypotenuse of the triangles in Square and also the side length of the square in the middle. Give it a try!

Two squares of the same area are labeled “F” and “G”. Square F is divided into the following: A square with side lengths “a”. Two rectangles with side lengths “a” and “b”. A square with side lengths “b”. Square G is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths.

There are many examples where the lengths of two legs of a right triangle are known and can be used to find the length of the hypotenuse with the Pythagorean Theorem. The Pythagorean Theorem can also be used if the length of the hypotenuse and one leg is known, and we want to find the length of the other leg. Here is a right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by .

Start with , make substitutions, and solve for the unknown value. Remember that represents the hypotenuse: the side opposite the right angle. For this triangle, the hypotenuse is 10.

A right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by the letter g.

Use estimation strategies to know that the length of the other leg is between 8 and 9 units, since 75 is between 64 and 81. A calculator with a square root function gives .