# Lesson 14

Proving the Pythagorean Theorem

## 14.1: Notice and Wonder: Variable Version (5 minutes)

### Warm-up

The purpose of this warm-up is for students to get familiar and comfortable with the diagram of an altitude drawn to the hypotenuse of a right triangle with side lengths labeled with variables, which will be useful when students write equations using these variables and manipulate them in a later activity. By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1).

### Launch

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. Don’t feel the need to wrap anything up, the next activity has students exploring this diagram further.

## 14.2: Prove Pythagoras Right (15 minutes)

### Activity

All the work students have done to make sense of the altitude to the hypotenuse in right triangles pays off in this activity, as students prove the familiar Pythagorean Theorem. In middle school, students explored and proved the Pythagorean Theorem visually, but now they can also prove it using similarity. Because manipulating the equivalent ratios involved in this proof is challenging, and it’s hard to see where it’s going when you’re in the middle of calculations, we introduce the proof through a dialogue between two characters, but students are the ones to come up with the insight of how we can use the expressions involving $$a^2$$ and $$b^2$$ to prove that $$a^2 + b^2 = c^2$$.

### Launch

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. Students may benefit from highlighting the two triangles in different colors so that they can discern which angles and sides are relevant, or drawing the two triangles separately and color coding the corresponding sides to create the proportions.
Supports accessibility for: Visual-spatial processing, conceptual processing

### Student Facing

Elena is playing with the equivalent ratios she wrote in the warm-up. She rewrites $$\frac{a}{x} = \frac{c}{a} \text{ as } a^2=xc$$. Diego notices and comments, “I got $$b^2=yc$$. The $$a^2$$ and $$b^2$$ remind me of the Pythagorean Theorem.” Elena says, “The Pythagorean Theorem says that $$a^2 + b^2 = c^2$$. I bet we could figure out how to show that.”

1. How did Elena get from $$\frac{a}{x} = \frac{c}{a} \text{ to } a^2=xc$$?
2. What equivalent ratios of side lengths did Diego use to get $$b^2=yc$$?
3. Prove $$a^2+b^2=c^2$$ in a right triangle with legs length $$a$$ and $$b$$ and hypotenuse length $$c$$.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students start their proof by substituting expressions into $$a^2+b^2=c^2$$ tell them great job working backwards! To write a proof they need to reorganize their ideas to start with things they know are true and end the proof with $$a^2+b^2=c^2$$.

### Activity Synthesis

Invite students to share their thinking about how Elena’s and Diego’s discoveries can be used to prove the Pythagorean Theorem. Continue discussing until students have described all the key ideas:

• adding the expressions equal to $$a^2$$ and $$b^2$$ together
• figuring out why $$xc + yc$$ = $$(x+y)c$$
• figuring out why $$(x+y)c = c^2$$.

Ask students if $$a^2 + b^2 = c^2$$ is true for all types of triangles. If students aren’t sure, display an example of an altitude in an acute triangle. Once students are convinced that $$a^2 + b^2 = c^2$$ only works for right triangles, ask students what aspect of the proof only worked for right triangles. (The altitude only forms three similar triangles if the biggest triangle is a right triangle.)

## 14.3: An Alternate Approach (15 minutes)

### Activity

Students may be familiar with this proof from middle school. Revisiting this proof helps students connect a familiar diagram and proof to the similarity proof. It also gives students an opportunity to compare and contrast different types of proof and think about why mathematicians might bother to re-prove something that has already been proved.

### Launch

Use the Notice and Wonder routine to help students make sense of the context and images they will be using. Ask students what they notice and wonder about the images. If not mentioned by students, ask where $$c$$ shows up in these images.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for how Pythagoras used the diagrams to prove $$a^2+b^2=c^2$$. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do you know that the two large squares are the same size?”, and “How do you know that the area of the blue square is $$c^2$$?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify the proof of the Pythagorean Theorem.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide access to the different pieces of the figures so students can rearrange and compare them.
Supports accessibility for: Visual-spatial processing; Conceptual processing

### Student Facing

When Pythagoras proved his theorem he used the 2 images shown here. Can you figure out how he used these diagrams to prove $$a^2+b^2=c^2$$ in a right triangle with hypotenuse length $$c$$?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

James Garfield, the 20th president, proved the Pythagorean Theorem in a different way.

• Cut out 2 congruent right triangles
• Label the long sides $$b$$, the short sides $$a$$ and the hypotenuses $$c$$.
• Align the triangles on a piece of paper, with one long side and one short side in a line. Draw the line connecting the other acute angles.

How does this diagram prove the Pythagorean Theorem?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Encourage students to look for $$a^2$$. Students have scissors and tracing paper in their geometry toolkits, they may want to trace, cut out, and reassemble the pieces.

### Activity Synthesis

Invite students to talk about the differences between this proof and the one in the previous activity, including which one made more sense to them, which one they preferred writing, or which one felt most convincing. Ask students why they think mathematicians might continue to try to come up with different ways to prove a theorem as old as the Pythagorean Theorem.

## Lesson Synthesis

### Lesson Synthesis

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Pythagorean Theorem: If a right triangle has legs with lengths $$a$$ and $$b$$
and hypotenuse with length $$c$$, then $$a^2 + b^2 = c^2$$.  (Theorem)

Use a Think Pair Share routine to have all students consider and say the converse of the Pythagorean Theorem. (In a triangle with side lengths $$a$$ and $$b$$ and longest side $$c$$, if $$a^2 + b^2 = c^2$$, then the triangle is a right triangle.)

Remind students that they probably proved the converse of the Pythagorean Theorem in middle school. Ask students to brainstorm the kinds of problems the Pythagorean Theorem helps them solve, and the kinds of problems its converse helps them solve.

## 14.4: Cool-down - Test it Out (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

In any right triangle with legs $$a$$ and $$b$$ and hypotenuse $$c$$, we know that $$a^2+b^2=c^2$$. We call this the Pythagorean Theorem. But why does it work?

We can use an altitude drawn to the hypotenuse of a right triangle to prove the Pythagorean Theorem.

We can use the Angle-Angle Triangle Similarity Theorem to show that all 3 triangles are similar. Because the triangles are similar, corresponding side lengths are in the same proportion.

Because the largest triangle is similar to the smaller triangle, $$\frac{c}{a}=\frac{a}{d}$$. Because the largest triangle is similar to the middle triangle, $$\frac{c}{b}=\frac{b}{e}$$. We can rewrite these equations as $$a^2=cd$$ and $$b^2=ce$$.

We can add the 2 equations to get that $$a^2+b^2=cd+ce$$ or $$a^2+b^2=c(d+e)$$. From the original diagram we can see that $$d+e=c$$, so $$a^2+b^2=c(c)$$ or $$a^2+b^2=c^2$$.

Using the Pythagorean Theorem we can describe a triangle's angles without ever drawing it. For example, a triangle with side lengths 8, 15, and 17 is right because $$17^2=8^2+15^2$$. A triangle with side lengths 8, 15, and 18 is obtuse because $$18^2>8^2+15^2$$. A triangle with side lengths 8, 15, and 16 is acute because $$16^2<8^2+15^2$$.