# Lesson 8

Are They All Similar?

## 8.1: Stretched or Distorted? Rectangles (10 minutes)

### Warm-up

In a previous lesson, students were reminded of the definition of similarity in terms of dilations and rigid transformations (which they first encountered in middle school). This warm-up gives students an opportunity to use that definition in an argument. Monitor for students who use an alternative definition of similarity as “same shape, different size,” producing arguments such as:

- they are similar because they’re both rectangles
- they aren’t similar because one is skinnier than the other
- they aren’t similar because pairs of corresponding side lengths aren’t all in the same proportion

Identify students using dilation-based arguments:

- they are similar because one is a stretched-out version of the other
- they aren’t similar because if you dilate based on the scale factor that makes these sides match, the other sides won’t match, and vice versa

### Launch

Save plenty of time for the synthesis of this activity. Students should not need more than 3 minutes to gather their thoughts, and then the remaining 7 minutes can be used to synthesize and activate students’ prior knowledge for use in later activities.

### Student Facing

Are these rectangles similar? Explain how you know.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Anticipated Misconceptions

If students are unsure where to start or argue that one figure is a dilation of the other because it’s stretched out, remind them of the distorted hippos from the beginning of this unit. If these rectangles had photographs inside them, would it be a proper enlargement or distort the image?

### Activity Synthesis

Invite students with both general and dilation-based arguments to share. Ask the class to translate the general arguments into dilation-based arguments.

For example, if a student said that the rectangles were not similar because one was much skinnier than the other, the class could revise the statement to say that there couldn’t be a dilation from one rectangle to the other because if you made the scale factor big enough to make the horizontal sides match, the dilated rectangle would be way too tall.

## 8.2: Faulty Logic (10 minutes)

### Activity

Tyler’s proof gives students a chance to see the structure of a similarity proof. In addition, it gives students a chance to do error analysis and provides another way to explain why not all rectangles are similar. Students should recognize many of the moves and justifications from proofs about congruence in a previous unit, and proofs about dilations earlier in this unit.

This activity also gives the class an opportunity to discuss the proof process. Sometimes students who struggle in geometry assume that they should just know if a statement is true, and try to jump to proof before they have explored. Mathematicians spend much of their time experimenting, wondering, guessing, sketching, and being unsure. Tyler’s mistake probably came from not experimenting and sketching before he began writing a proof. In subsequent activities, students will benefit from exploring and thinking about whether each statement is true before trying to prove it.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Emphasize to students that they should draw exactly what the proof says for every step. Encourage students to draw different rectangles for \(ABCD\) and \(PQRS\) that will help them prove that Tyler’s statement really is false.

*Representation: Internalize Comprehension.*Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.

*Supports accessibility for: Language; Conceptual processing*

### Student Facing

Tyler wrote a proof that all rectangles are similar. Make the image Tyler describes in each step in his proof. Which step makes a false assumption? Why is it false?

- Draw 2 rectangles. Label one \(ABCD\) and the other \(PQRS\).
- Translate rectangle \(ABCD\) by the directed line segment from \(A\) to \(P\). \(A’\) and \(P\) now coincide. The points coincide because that’s how we defined our translation.
- Rotate rectangle \(A’B’C’D’\) by angle \(D’A’S\). Segment \(A’’D’’\) now lies on ray \(PS\). The rays coincide because that’s how we defined our rotation.
- Dilate rectangle \(A’’B’’C’’D’’\) using center \(A’’\) and scale factor \(\frac{PS}{AD}\). Segments \(A’’’D’’’\) and \(PS\) now coincide. The segments coincide because \(A’’\) was the center of the rotation, so \(A’’\) and \(P\) don’t move, and since \(D’’\) and \(S\) are on the same ray from \(A’’\), when we dilate \(D’’\) by the right scale factor, it will stay on ray \(PS\) but be the same distance from \(A’’\) as \(S\) is, so \(S\) and \(D’’’\) will coincide.
- Because all angles of a rectangle are right angles, segment \(A’’’B’’’\) now lies on ray \(PQ\). This is because the rays are on the same side of \(PS\) and make the same angle with it. (If \(A’’’B’’’\) and \(PQ\) don’t coincide, reflect across \(PS\) so that the rays are on the same side of \(PS\).)
- Dilate rectangle \(A’’’B’’’C’’’D’’’\) using center \(A’’’\) and scale factor \(\frac{PQ}{AB}\). Segments \(A’’’’B’’’’\) and \(PQ\) now coincide by the same reasoning as in step 4.
- Due to the symmetry of a rectangle, if 2 rectangles coincide on 2 sides, they must coincide on all sides.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

Begin the discussion by inviting students to explain what Tyler did well, and what makes a good proof that two figures are similar. (Tyler gave reasons why each of his transformations worked. Tyler used rigid transformations and dilations.) Then, discuss where Tyler went wrong. (When Tyler did the second dilation, it changed the lengths from the first dilation, so the first pair of corresponding sides aren’t congruent anymore.)

Ask students what Tyler could have done before he even wrote the proof to make sure he didn’t waste time proving something that wasn’t true. Help students see that experimenting, looking for examples and counterexamples, and drawing pictures is part of the proof process.

## 8.3: Always? Prove it! (15 minutes)

### Activity

After conjecturing, viewing, and critiquing examples of a similarity proof using rigid transformations and dilations, students are ready to write proofs of the two true statements in this activity. Monitor for students who include any of these points in their proof that all circles are similar:

- there will always be a dilation that makes the circles have the same radii
- there will always be a rigid motion that takes one circle’s center onto the other
- two circles with the same center and the same radius are the same circle (coincide)

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Remind students of the importance of drawing pictures and experimenting before trying to prove.

*Representing, Conversing: MLR7 Compare and Connect.*Use this routine to prepare students for the whole-class discussion about the proofs that all equilateral triangles are similar and all circles are similar. After students write a proof for either all equilateral triangles or all circles are similar, 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 proofs. 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 sequence of transformations used in each proof.

*Design Principle(s): Cultivate conversation*

*Representation: Internalize Comprehension.*Activate or supply background knowledge. Provide students with access to definitions with examples of equilateral triangles, isosceles triangles, right triangles and circles.

*Supports accessibility for: Visual-spatial processing; Organization*

### Student Facing

Choose one statement from the list. Decide if it is true or not.

If it is true, write a proof. If it is not, provide a counterexample.

Repeat with another statement.

Statements:

- All equilateral triangles are similar.
- All isosceles triangles are similar.
- All right triangles are similar.
- All circles are similar.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Student Facing

#### Are you ready for more?

Here is an \(x\) by \(x+1\) rectangle and a 1 by \(x\) rectangle. They are similar. What are the possible dimensions of these golden rectangles? Explain or show your reasoning.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

### Anticipated Misconceptions

If students are stuck writing their proof, suggest they use the model from the previous activity (the structure is valid despite the error in reasoning).

### Activity Synthesis

Begin with the statements that students were able to prove false. Invite students to share examples and counterexamples. Ask students if having an example of a statement makes it true. (No, statements are only true if they are always true for all examples. One counterexample is enough to prove a statement false.)

Invite several students to share their reasoning about circles. Their explanations should include:

- there will always be a dilation that makes the circles have the same radii
- there will always be a rigid motion that takes one circle’s center onto the other
- two circles with the same center and the same radius are the same circle (coincide)

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

All circles are similar. (Theorem)

## Lesson Synthesis

### Lesson Synthesis

Invite students to consider and connect two questions:

- What is it about rectangles that means that they are not all similar to each other, while all equilateral triangles are similar to one another?
- How do you prove rigorously that a whole set of shapes are similar to one another?

Invite a student to display or share their proof all equilateral triangles are similar. Ask students to discuss why the proof that all equilateral triangles are similar doesn’t use the same faulty logic that Tyler used proving all rectangles are similar.

If students struggle to make connections, here are some intermediate questions:

- What went wrong in Tyler’s proof? (He tried to do two dilations and the second one messed up the first one.)
- Could someone make that same mistake in another proof? (Sure, if they try to do two different dilations. The second dilation, unless the scale factor is 1, will change all the lengths.)
- Did we accidentally make that mistake in our proof that all equilateral triangles are similar? (No, we only had to use one dilation.)
- Does it work to only use one dilation in our proof? Are you sure we shouldn’t have done one dilation to get the first pair of sides to line up and another dilation to get the second pair of sides to line up? (No, the same dilation works for all three sides, because all the sides in an equilateral triangle are congruent to each other, so the same scale factor will work for all three pairs of corresponding sides.)
- Why did Tyler try to use two dilations? (If you need to use two scale factors, that is a sign that the figures aren’t really similar. He needed one dilation for one dimension of the rectangle and another dilation for the other dimension, but having two different scale factors means the figures aren’t similar.)

## 8.4: Cool-down - Samesies (5 minutes)

### Cool-Down

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs.

## Student Lesson Summary

### Student Facing

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. Consider any 2 circles, \(A\) and \(B\). Translate the circle centered at \(A\) along directed line segment \(AB\).

Now a dilation with center \(B\) and a scale factor that is the length of the radius of the circle centered at \(B\) divided by the length of the radius of the circle centered at \(A\) will take the circle centered at \(A\) onto the circle centered at \(B\), proving that all circles are similar.

We can also show that all equilateral triangles are similar. Because we are talking about triangles, we can use the theorem that having all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion is enough to prove that the triangles are similar. All the pairs of corresponding angles are congruent because all the angles in both triangles measure \(60^{\circ}\). All the pairs of corresponding side lengths must be in the same proportion, because within each triangle, all the sides are congruent. Therefore, whatever scale factor works for one pair of sides will work for all 3 pairs of corresponding sides. If all pairs of corresponding sides are in the same proportion and all pairs of corresponding angles are congruent, then all equilateral triangles are similar.