Lesson 6

In a subsequent lesson, students will justify why not all rectangles are similar to each other, and explore a false proof that all rectangles are similar. This activity previews that thinking, as students explain what went wrong with this attempt to dilate a square. Monitor for students who:

• notice the figures are not the same shape
• notice the sides are not scaled by the same scale factor
• notice that \(E\) and \(F\) are not on rays \(PB\) and \(PC\)

The goal of this synthesis is for students to begin to justify why accurate dilations lead to figures with all pairs of corresponding angles congruent and all pairs of corresponding sides having the same ratio as the scale factor. They approach this from the contrapositive: if a figure does not have all pairs of corresponding angles congruent and all pairs of corresponding sides having the same ratio as the scale factor, then the figures can’t be a dilation of one another.

Invite students to share who:

• noticed the figures are not the same shape
• noticed the sides are not scaled by the same scale factor
• noticed that \(E\) and \(F\) are not on rays \(PB\) and \(PC\)

If scale factor or angles are not mentioned by students, ask, “All the angles are the same, doesn’t that mean they’re similar?” (Angles need to be congruent and sides need to be in the same ratio.) If no student noticed \(E\) and \(F\) are not on rays \(PB\) and \(PC\), draw in the auxiliary lines. The goal is for students to specify the properties that must be true if a dilation is to exist between two figures.

The goal of this card sort is to surface the connections among dilations and rigid transformations as well as similarity and congruence. After doing and discussing a card sort, students are assigned one pair of figures to figure out a sequence of rigid motions and dilations that takes one figure onto the other. Students use their prior knowledge of rigid transformations and their experience carrying out dilations to help them be successful in this task.

Monitor for different ways groups choose to categorize the pairs of figures:

• figures that are similar 
• figures that are congruent
• figures that are neither congruent nor similar 

Listen for reasons that students provide. Do they use language of similarity, dilation, or same shape?

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the pairs of figures, they should work with their partner to come up with categories.

After students have sorted their cards, assign each group a card to write transformations. Do not assign the rhombi to any groups.

If students struggle to describe their transformations, remind them they can use tracing paper to redraw the figures and add labels to the points.

Select groups to share their categories. Ensure that at least one set of categories distinguishes between: 

• figures that are similar 
• figures that are congruent
• figures that are neither congruent nor similar 

Record the scale factors for dilating from Figure \(F\) to Figure \(G\), and Figure \(G\) to Figure \(F\) for each pair of figures except the rhombi. Include the right triangles and the trapezoids, which have a scale factor of 1. Ask students what they notice. (There’s a scale factor either way. The scale factors are multiplicative inverses. The congruent figures have a scale factor of 1.)

The goal of this activity is for students to use the definition of similarity, and to practice applying some of the skills they learned when studying congruent figures to similar figures. Students practice writing a sequence of rigid motions and dilations to take one figure onto another, write accurate similarity statements about the figures, and reason about what must be true about corresponding side lengths.

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

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. (Definition)

Translation and dilation takes \(\triangle ABC\) onto \(\triangle FDE\) so \(\triangle ABC \sim \triangle FDE\)

Tell students \(\triangle ABC \sim \triangle FDE\) is a similarity statement, and we read \(\sim\) as “is similar to.”

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion. 

If students struggle to define sequences of rigid motions and dilations, review an example from the card sort. 

Ask students to determine if their sequence of rigid motions and dilations will always work for any pair of triangles that have all pairs of corresponding angles congruent and all pairs of corresponding sides lengths in the same proportion. (Probably not, some figures would require rotations and dilations. But the general method of dilating and using rigid motions will always work.)

For students who dilated first, ask:

• How do you choose the dilation to use? (Dilate from any vertex, by the scale factor given by the ratio of a pair of corresponding sides.)
• Once you dilate, how do you know you’ll be able to use rigid motions to take one triangle to the other? (The triangles will be congruent, because their corresponding angles will remain congruent, and all their side lengths will be multiplied by the value \(k\) that you need to make them the same length as the corresponding side. Since all the corresponding angles and all the corresponding sides will match, you’ll be able to take one triangle to the other.)

For students who used rigid motions first, ask:

• How do you know rigid motion will line up parts of the figures? (Because corresponding angles are congruent, we’ll be able to line up both of the rays that meet at the vertex by translating and rotating, because you can always take an angle onto another congruent angle.)
• How do you know that dilation will then line up the remaining two points? (The sides lengths will be in the same proportion, because we’re given that information, and the angles all line up, so the triangles are dilations of each other.)