# Lesson 13

Using the Pythagorean Theorem and Similarity

## 13.1: Similar, Right? (5 minutes)

### Warm-up

The goal of this activity is to get students familiar with the two smaller right triangles formed by drawing an altitude to the hypotenuse of a right triangle. The activity previews the activities that students will do later in this lesson and the next. Listen to hear if students compare the two smaller triangles to the larger triangle, or make conjectures about what other proportional relationships might be present.

### Student Facing

Is triangle $$ADC$$ similar to triangle $$CDB$$? Explain or show your reasoning.

### Anticipated Misconceptions

If students struggle to see which angles and sides correspond, encourage them to copy each triangle onto tracing paper and rotate them so they all have the same orientation. Colored pencils can also help students identify corresponding parts.

### Activity Synthesis

Ask students how many triangles they see in the diagram. Ask them how many of the triangles are similar. If students aren’t sure, they will have more opportunities later in the lesson to untangle the diagram.

Tell students an altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.

## 13.2: Tangled Triangles (15 minutes)

### Activity

In a subsequent lesson, students will use proportional relationships in a similar diagram to prove the Pythagorean Theorem. It’s important for students to practice seeing the relationships between the triangles formed by drawing an altitude to the hypotenuse of a right triangle. Monitor for students who:

• use the Pythagorean Theorem to find the third side length
• use the scale factor they determined to find the third side length
• use equivalent ratios to find the third side length

### Launch

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion about strategies for calculating the lengths of sides $$HG$$, $$GF$$, and $$HF$$. After students calculate the side lengths of each triangle, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the various strategies for calculating the side lengths of similar right triangles.
Design Principle(s): Cultivate conversation
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. Students may benefit from highlighting the corresponding sides and angles so that they can see the relationships better.
Supports accessibility for: Visual-spatial processing, conceptual processing

### Student Facing

Trace the 2 smaller triangles onto separate pieces of tracing paper.

1. Turn your tracing paper and convince yourself all 3 triangles are similar.
2. Write 3 similarity statements.
3. Determine the scale factor for each pair of triangles.
4. Determine the lengths of sides $$HG$$, $$GF$$, and $$HF$$.

### Anticipated Misconceptions

If students struggle to find the similarity statement, have them also trace the large triangle without the altitude and then have them orient all three triangles in the same direction.

### Activity Synthesis

Invite a student to share who:

• used the Pythagorean Theorem to find the third side length
• used the scale factor they determined to find the third side length
• used equivalent ratios to find the third side length

If no student used equivalent ratios, ask students to write several equivalent ratios for the three triangles. In a subsequent lesson, students will need to recognize, write, and manipulate equivalent ratios involving the same side lengths.

## 13.3: More Tangled Triangles (15 minutes)

### Activity

In this activity, the angle measures are not given, so students need to convince themselves that the triangles in question are similar. Students may be tempted to assume they are similar simply because of the triangles in the previous diagram that looked like this, so monitor for students actively trying to convince themselves that the Angle-Angle Triangle Similarity Theorem applies in this case. Look for:

• students measuring angles (with tracing paper or protractors)
• students measuring one or two angles and calculating the rest
• students using algebraic or logical reasoning to convince themselves that the angles would be congruent

Students also get another chance to write equivalent ratios. One key thing they might notice is that a lot of the sides are in more than one ratio.

### Launch

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to increase awareness of the language used to talk about the features of similar right triangles. Before revealing the questions in this activity, display the image of the triangles formed by drawing an altitude to the hypotenuse of right triangle $$ABC$$. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of all three triangles. For example, “Are all three right triangles similar?”, “What is the length of $$BC$$?”, and “What angles are congruent to angle $$B$$?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “_______________ reminds me of __________ because . . .”,  “_________ will always ____________ because . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

1. Convince yourself there are 3 similar triangles. Write a similarity statement for the 3 triangles.
2. Write as many equations about proportional side lengths as you can.
3. What do you notice about these equations?

### Student Facing

#### Are you ready for more?

Tyler says that since triangle $$ACD$$ is similar to triangle $$ABC$$, the length of $$CB$$ is 11.96. Noah says that since $$ABC$$ is a right triangle, we can use the Pythagorean Theorem. So the length of $$CB$$ is 12 exactly. Do you agree with either of them? Explain or show your reasoning.

### Anticipated Misconceptions

If students assume the triangles are similar because the diagram looks like the previous diagrams, ask them how they know.

### Activity Synthesis

Invite students to explain how they convinced themselves that triangles $$ABC, ACD,$$ and $$CBD$$ are similar. Select students to share their thinking in this order:

• students who used tracing paper or measured all the angles
• students who measured just one or two angles and calculated the rest
• students who used algebraic or logical reasoning to convince themselves that the angles would be congruent

After students who traced or measured share, make sure all students understand and appreciate the idea that it’s necessary to know that the triangles are similar before considering missing side lengths and that finding angle measures is important to establish the triangles are similar.

Finally, ask students to share equivalent ratios that they found. Record equivalent ratios in which the same side appears twice in a separate column, for example $$\frac{AD}{DC} = \frac{DC}{DB}$$ and $$\frac{CB}{AB} = \frac{DB}{CB}$$. Ask students what they notice about the equivalent ratios grouped together.

## Lesson Synthesis

### Lesson Synthesis

Ask students to find the measures of all of the missing angles and justify how they know. (Angle $$BCD$$ measures $$58^{\circ}$$ by the Triangle Angle Sum Theorem. Angle $$ACD$$ measures $$32^{\circ}$$ because it is the complement of angle $$BCD$$. Angle $$A$$ measures $$58^{\circ}$$ by the Triangle Angle Sum Theorem.)

Repeat with a variable in place of the $$32^{\circ}$$ angle. (Angle $$BCD$$ measures $$x^{\circ}$$. Angle $$ACD$$ measures $$90^{\circ} - x^{\circ}$$. Angle $$A$$ measures $$x^{\circ}$$.) Once the angles are calculated, display for all to see the triangles redrawn so they are all oriented the same way. Invite multiple students to explain why the triangles formed by drawing the altitude to the hypotenuse must be similar. Encourage students to generalize their conclusions to any right triangle with the altitude drawn to the hypotenuse. Ask students to list all of the equivalent ratios that they can among the three similar triangles. $$\left(\frac{AD}{CA} = \frac{CD}{CB} = \frac{CA}{AB}, \frac{CD}{CA} = \frac{DB}{BC} = \frac{BC}{BA} \right)$$Point out again the equivalent ratios in which one side appears twice in the same equation. Students will need to use these in a subsequent lesson to prove the Pythagorean Theorem.

Remind students that these triangles are no different than any other set of similar triangles. All the same strategies they have for reasoning about similar triangles still apply to this diagram.

## Student Lesson Summary

### Student Facing

When we draw an altitude from the hypotenuse of a right triangle, we get lots of similar triangles that can be used to find missing lengths. An altitude is a segment from one vertex of the triangle to the line containing the opposite side that is perpendicular to the opposite side. For right triangle $$PQR$$ we can draw the altitude $$PS$$

Why are triangles $$PQR$$, $$SQP$$, and $$SPR$$ all similar to each other?
Triangles $$PQR$$ and $$SQP$$ are similar by the Angle-Angle Triangle Similarity Theorem because angle $$Q$$ is in both triangles, and both triangles are right triangles, so angles $$RPQ$$ and $$PSQ$$ are congruent. Triangles $$PQR$$ and $$SPR$$ are similar by the Angle-Angle Triangle Similarity Theorem because angle $$R$$ is in both triangles, and both triangles are right triangles, so angles $$RPQ$$ and $$RSP$$ are congruent. Because triangles $$SQP$$ and $$SPR$$ are both similar to triangle $$PQR$$, they are also similar to each other.
Since the triangles $$PQR$$, $$SQP$$, and $$SPR$$ are all similar, corresponding angles are congruent and pairs of corresponding sides are scaled copies of each other, by the same scale factor. We can use the proportionality of pairs of corresponding side lengths to find missing side lengths. For example, suppose we need to find $$PS$$ and know $$RS=3$$ and $$QS=7$$. Since triangle $$SQP$$ is similar to triangle $$SPR$$, we know $$\frac{RS}{PS}=\frac{PS}{QS}$$. So $$\frac{3}{PS}=\frac{PS}{7}$$ and $$PS=\sqrt{21}$$. Or, suppose we need to find $$SQ$$ and know $$PQ=5$$ and $$RQ=12$$. Since triangle $$PQR$$ is similar to triangle $$SQP$$, we know $$\frac{RQ}{PQ}=\frac{PQ}{SQ}$$. So $$\frac{12}{5}=\frac{5}{SQ}$$ and $$SQ=\frac{25}{12}$$.