Unit 2, Lesson 14
Side Length Quotients in Similar Triangles
Accelerated 7
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Geometry toolkits
In this activity, students identify whether two triangles are similar or not. Since no drawing is given, students will need to recognize that there is no single scale factor that multiplies all of the side lengths in one triangle to get the side lengths in the other triangle.
Provide access to geometry toolkits. Give students 2 minutes of quiet work time followed by a whole-class discussion.
Triangle A has side lengths 2, 3, and 4. Triangle B has side lengths 4, 5, and 6.
Is Triangle A similar to Triangle B? Be prepared to explain your reasoning.
Some students may think that adding the same number to each side length will result in similar triangles. Draw a picture to help students see why this is not true.
The goal of this discussion is to make sure students understand that triangles cannot be similar if you cannot apply the same scale factor to each side of one triangle to get the corresponding sides of the other triangle. Discuss with students:
Display diagrams of the triangles for visual confirmation.
MLR8: Discussion Supports
None
While previous activities explored the ratio of side lengths between similar triangles, this activity explores ratios of side lengths within similar triangles. and how these compare for similar triangles. For example, if
Arrange students in groups of 3. Assign each of the columns in the second table to 1 student in each group. Give students 5 minutes of quiet work time followed by a partner then whole-class discussion.
Triangle
The scale factors for the dilations that show triangle
| triangle | scale factor | length of short side |
length of medium side |
length of long side |
|---|---|---|---|---|
| 1 | 4 | 5 | 7 | |
| 2 | ||||
| 3 | ||||
| triangle | (long side) |
(long side) |
(medium side) |
|---|---|---|---|
|
|
|
|
|
What do you notice about the quotients?
The goal of this discussion is to make sure students understand that quotients of corresponding side lengths in similar triangles are equivalent. Discuss with students:
“For the triangles examined, what would the value of (medium side)
“Do you think the value of (medium side)
MLR7: Compare and Connect
None
In this activity, students calculate side lengths of similar triangles. They need to think strategically about which side lengths to calculate first since there are many missing values. As they discover more side lengths, more paths for finding the remaining values open up.
Monitor for students who use these different strategies:
Uses (external) scale factors to move from one triangle to another
Uses quotients of corresponding side lengths within a triangle (internal scale factors)
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by a partner then whole-class discussion.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Triangles
The side lengths of the triangles all have the same units. Find the unknown side lengths.
Three triangles. Vertices capital A, B C. Sides B C and A, C length 4. Side A, B length lower-case c. Vertices capital I H G. Side I G length lower-case h. Side I H length fraction 12 over 5. Side H G length fraction 6 over 5. Vertices capital D E F. Side E F 5, Side F D length lower-case e, side E D length lowercase d.
The goal of this discussion is to emphasize how multiple relationships can be used to find side lengths of similar triangles. Display 2–3 strategies from previously selected students for all to see. If time allows, invite students to briefly describe their strategies. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“What do the different strategies have in common? How are they different?”
“Did anyone solve the problem the same way, but would explain it differently?”
“How does each method represent scale factor?”
“Are there any benefits or drawbacks to one method compared to another?”
Both methods are efficient and the method to use is guided by what information is missing and the numbers involved in the calculations. Some key points to highlight are:
Triangle
One side of triangle
The purpose of this discussion is to compare the two different ways for finding the side lengths of similar triangles—using scale factors to to move from one triangle to another and using quotients of corresponding side lengths within the triangles. Display the image for all to see.
Tell students that these 2 triangles are similar, and since sides
Sometimes both methods for calculating missing side lengths are equally effective. Invite students to use one of these strategies to calculate the length of
If 2 polygons are similar, then the side lengths in one polygon are multiplied by the same scale factor to give the corresponding side lengths in the other polygon.
For these triangles the scale factor is 2:
Here is a table that shows relationships between the lengths of the short and medium sides of the 2 triangles.
| small triangle | large triangle | |
|---|---|---|
| medium side | 4 | 8 |
| short side | 3 | 6 |
| (medium side) |
The lengths of the medium side and the short side are in a ratio of
We can use these facts to calculate missing lengths in similar polygons. For example, triangles
Since side
Help us improve by sharing suggestions or reporting issues.
Some students may find quotients in fraction form. If necessary, prompt them to express their fractions in a way that helps them to recognize that the fractions are equivalent.
Some students may have trouble locating corresponding sides. Suggest they use tracing paper to rotate and or translate the triangles. Another technique is to color corresponding side lengths the same color. For example, they could color