# Lesson 15

Finding All the Unknown Values in Triangles

## 15.1: Which One Doesn’t Belong: Triangles (5 minutes)

### Warm-up

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

### Student Facing

Which one doesn’t belong?

### Student Response

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### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use. Also, press students on unsubstantiated claims.

## 15.2: Info Gap: Similar Sequence (20 minutes)

### Activity

This info gap activity gives students an opportunity to determine and request the information needed to calculate side lengths of similar right triangles. This activity previews the trigonometry concepts students will use in the next lesson. Students have enough information about one triangle to determine the ratios of any pair of sides within it, and to use the ratios to find missing side lengths in the other triangles. This is the work students will do in trigonometry, where they will learn to look up the information about a triangle from a trigonometry table or using calculating technology rather than being given it in a diagram.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Monitor for students using:

- a scale factor between triangles (\(XY = \frac{2}{3} PQ\))
- equivalent ratios between triangles (\(\frac{XY}{PQ} = \frac{YZ}{QR}\))
- equivalent ratios within triangles (\(\frac{XZ}{ZY} = \frac{PR}{RQ}\))

### Launch

Tell students they will be calculating side lengths of similar triangles. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

*Conversing:*This activity uses

*MLR4 Information Gap*to give students a purpose for discussing information necessary to solve problems involve calculating the side lengths of similar right triangles. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"

*Design Principle(s): Cultivate Conversation*

*Engagement: Develop Effort and Persistence.*Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.

*Supports accessibility for: Memory; Organization*

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

- Silently read the information on your card.
- Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
- Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
- Read the problem card, and solve the problem independently.
- Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

- Silently read your card and think about what information you need to answer the question.
- Ask your partner for the specific information that you need.
- Explain to your partner how you are using the information to solve the problem.
- When you have enough information, share the problem card with your partner, and solve the problem independently.
- Read the data card, and discuss your reasoning.

### Student Response

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### Anticipated Misconceptions

If students are stuck at first, suggest that asking for the similarity statement would be a good place to start. Then they can draw a diagram and figure out what other information they might want to ask for.

If students are stuck after asking for the available side lengths, ask what type of triangle would allow them to calculate the third side. (A right triangle.)

### Activity Synthesis

Ask students when the Pythagorean Theorem works. (Only with right triangles.) Remind students to always check that a triangle has a right angle before using the Pythagorean Theorem.

Invite a pair to share who:

- used the scale factor between triangles (either by asking for it or figuring it out).
- used the ratio within one triangle to determine the sides of the other triangle (either by creating a scale factor or using equivalent ratios)

Both methods work and one might be easier than the other depending on given information. In a subsequent unit students will learn trigonometry which relies on within triangle ratios.

## 15.3: Relatively Reasonable (10 minutes)

### Activity

This activity gives students a chance to recall all of the relationships within similar right triangles they know to be true, and attempt to use those relationships creatively to best estimate the side lengths of the unknown triangle. Monitor for students:

- estimating the length of one side
- using tracing paper and other materials to help with the estimation
- estimating a scale factor or establishing a scale factor based on an estimated side
- using the Pythagorean theorem

### Launch

Demonstrate how *not* to do this activity: Compare each side to its corresponding side using some gesturing and squinting, and announce after about 3 seconds that you’re done and that the sides are 5, 6, 7. Call on students to explain why that approach is not sufficient, what other clues they could use to check their thinking, and why 5, 6, 7 definitely can’t be the correct lengths of the small triangle. (5, 6, 7 doesn't work because \(\frac56 \neq \frac58\) so the sides aren't proportional.)

*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 triangles \(ABC\) and \(A’B’C’\). 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 both triangles. For example, “What is the scale factor between the triangles?”, “What is the length of \(BC\)?”, and “What is the length of \(B’C’\)?”

*Design Principle(s): Maximize meta-awareness; Support sense-making*

### Student Facing

### Student Response

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### Student Facing

#### Are you ready for more?

Find or make a square piece of paper. Fold the bottom left corner to the midpoint of the top edge. Label a point \(F\) at the midpoint of the segment created on the right edge, \(BE\). Prove \(F\) is \(\frac13\) of the way down the whole side of the square.

### Student Response

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### Activity Synthesis

Invite students to share all the different tools they used in their estimation. If possible, surface each of these tools:

- estimating the length of one side
- using tracing paper and other materials to help with the estimation
- estimating a scale factor or establishing a scale factor based on an estimated side
- using the Pythagorean theorem

## Lesson Synthesis

### Lesson Synthesis

“When finding all the unknown values in similar right triangles you have choices: you can use proportional relationships or the Pythagorean Theorem. Which would you use for each of these examples, and why?”

(The larger triangle is a scaled copy of the smaller one with a scale factor of 2, so it’s easy to find the missing lengths by multiplying the corresponding lengths by 2, so \(ED\) is 6 and \(FD\) is 8.)

(\(OMN\) is a right triangle so I can use the Pythagorean Theorem to find the missing length. That’s easier than finding the scale factor between 8.45 and 13. \(OM\) is 12. Finding \(JK\) will be a pain either way, but I’ll use the scale factor from 13 to 8.45 and apply that to \(MN\) and \(JK\). \(JK\) is 3.25.)

After students choose and defend their thinking, encourage them to check each other’s work using another method. Why should both methods give the same answer? (The side lengths don’t change just because how we calculate them does.)

## 15.4: Cool-down - Calculate and Check (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Since triangle \(ABC\) is a right triangle, \(4^2+x^2=5^2\). Since triangle \(FGH\) is a right triangle, \(y^2+\left(\frac{12}{5}\right)^2=4^2\). Since triangle \(ABC\) is similar to triangle \(FGH\), there are many equations to write using proportional relationships. We can use any combination of these equations to solve for \(x\) and \(y\).

By similarity, \( \frac 54 (y) = 4 \text{ so } y= \frac{16}{5}\). Substituting \(y= \frac{16}{5}\) into the Pythagorean Theorem gives \(\left( \frac{16}{5} \right)^2 + \left(\frac{12}{5}\right)^2=4^2\) which is true.

By the Pythagorean Theorem, \(x^2=5^2-4^2=9\) so \(x=3\). By similarity \(x = \frac{12}{5} \boldcdot \frac 54 \), which also equals 3.