Lesson 6 Pythagoras by Proportions Practice Understanding

Jump Start

In the diagram, the length of each side of the outer square is $a + b$ , and the length of each side of the inner square is $c$ . You are going to use this diagram to prove the Pythagorean theorem algebraically by showing that the area of the inner square, $c^{2}$ , is also equal to $a^{2} + b^{2}$ .

1.

Find an expression for the area of the outer square by expanding ${(a + b)}^{2}$ .

2.

Represent the area of one of the four triangles in terms of $a$ and $b$ .

3.

Find the area enclosed by the four small triangles in terms of $a$ and $b$ .

4.

Subtract the area of the four small triangles from the area of the larger square.

5.

How does this prove the Pythagorean theorem?

Learning Focus

Prove the Pythagorean theorem algebraically.

When and how do I use algebra in a geometric proof?

What does each proof of the Pythagorean theorem reveal?

Open Up the Math: Launch, Explore, Discuss

1.

There are many different proofs of the Pythagorean theorem. Here is one based on similar triangles.

Step 1: Cut a $4 \times 6$ index card along one of its diagonals to form two congruent right triangles.

Step 2: In each right triangle, draw an altitude from the right angle vertex to the hypotenuse. (Use the right angle in the other triangle to help you draw this altitude accurately.) Draw this altitude on both the front and back of the triangle.

Step 3: Label each triangle as shown in the diagram. Label the length of the altitude $h$ . Flip each triangle over and label the matching sides and angles with the same names on the back as on the front.

Step 4: Cut one of the right triangles along the altitude to form two smaller right triangles.

2.

Step 5: Arrange the three triangles in a way that convinces you that all three right triangles are similar. You may need to reflect and/or rotate one or more triangles to form this arrangement.

Step 6: Write proportionality statements to represent relationships between the labeled sides of the triangles. (Note: Side $c$ has been decomposed into segments labeled $x$ and $y$ . The sum of these two segments is $c$ .)

Step 7: Solve one of your proportions for $x$ and the other proportion for $y$ . (If you have not written proportions that involve $x$ and $y$ , study your set of triangles until you can do so.)

Step 8: Work with the equations you wrote in step 7 until you can show algebraically that $a^{2} + b^{2} = c^{2}$ . (Remember, $x + y = c$ .)

Ready for More?

Use your set of triangles to help you prove the following two theorems:

1.

Right triangle altitude theorem 1: If an altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the lengths of the two segments formed on the hypotenuse.

2.

Right triangle altitude theorem 2: If an altitude is drawn to the hypotenuse of a right triangle, the length of each leg of the right triangle is the geometric mean between the length of the hypotenuse and the length of the segment on the hypotenuse adjacent to the leg.

3.

Use your set of triangles to help you find the values of $x$ and $y$ in the provided diagram.

Takeaways

Here are some interesting things I noticed about right triangles today, by drawing the altitude of the triangle from the vertex at the right angle to the hypotenuse:

Vocabulary

altitude
geometric mean
Bold terms are new in this lesson.

Lesson Summary

In today’s lesson, we learned that drawing the altitude of a right triangle from the vertex at the right angle to the hypotenuse divides the right triangle into two smaller triangles that are similar to each other and to the original right triangle. We were able to prove the Pythagorean theorem using proportionality statements about the three similar triangles.

Retrieval

1.

What are the two ways to determine if two figures are similar?

2.

Which of the following are similar to each other? Why?

3.

An arithmetic sequence is represented in the table below. Find the missing values and write an explicit function rule for the sequence.

Term	$1$	$2$	$3$	$4$
Value	$3$			$24$

Explicit function rule:

4.

A geometric sequence is represented in the table below. Find the missing values and write an explicit function rule for the sequence.

Term	$1$	$2$	$3$	$4$
Value	$3$			$192$

Explicit function rule: