Similar Triangles

Standard(s) for Mathematical Practice (MP)

MP 1 Make sense of problems and persevere in solving them.
MP 5 Use appropriate tools strategically.
MP 6 Attend to precision.
MP 7 Look for and make use of structure.

Grade Level

Grades 8–10

Content Domain

Geometry Similarity, Right Triangles, and Trigonometry (Geometry Conceptual Category)

Standard(s) for Mathematical Content

8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Math Topic Keywords

  • similar triangles
  • similarity
  • transformations
  • rotations
  • reflections

The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task. Before reading the dialogue, work on the mathematics task. Next reflect on the mathematical practices you engaged in while working on the task. Finally read the student dialogue.

Mathematics Task

In the figure below, the two triangles are similar and points $A$, $C$, and $D$ are collinear and points $E$, $C$, and $B$ are collinear. What is the length of side $CD$?

Student Dialogue

Students in this dialogue have experience determining congruence of figures through translations, rotations, and reflections. They know that proportional relationships exist between corresponding sides of similar figures, and students are using geometric transformations to help them find the corresponding sides of similar figures in order to use their proportionality.

(1) Lee I know how to do this. They told us the two triangles are similar so we know the corresponding sides need to be proportional.
(2) Chris So, that means we need to figure out the scale factor, right?
(3) Lee Sure, but first we need to know which sides are corresponding.
(4) Matei Ok. Let’s figure out what corresponds to what then.
(5) Chris Will the corresponding sides be lined up if we flip the top triangle down?
(6) Matei Do you mean reflecting it over $C$? I mean, over the horizontal line through $C$? I don’t think that works. I think the corresponding sides won’t line up.
(7) Lee Let’s try it. [Lee traces the entire figure onto a piece of tracing paper, and folds the paper so that the vertical angles, $\angle{ACB}$ and $\angle{ECD}$, coincide.] Wait, so is this what you mean by “flip down?”

(8) Chris Yeah, that’s what I meant. You reflected the triangle. I guess it’s not over the horizontal line through $C$, but it’s a reflection and…
(9) Matei Okay, so see… The sides that are now lined up with each other are not corresponding sides.
(10) Chris How do you know?
(11) Matei Well, for one thing, these bottom lines, $AB$ and $ED$, are not parallel and I think they should be.
(12) Chris Is that enough to know the overlapping sides aren’t corresponding, though?
(13) Lee We could check to see if their side lengths are proportional. The question is whether the ratios of $EC$ to $AC$ and $ED$ to $AB$ are the same. [writes the following] $$ \begin{align*} \frac{8}{6}=1.\overline{3} && \frac{12.8}{8}=1.6 \end{align*}$$ These pairs of sides aren’t proportional so they can’t be corresponding!
(14) Chris What if we rotated the diagram instead of reflecting it. Would that help?
(15) Lee Oh, I like that. I’ll try it. [Lee unfolds the tracing, places it on top of the original diagram, and then rotates it $180$ degrees around point $C$.]

(16) Matei Look! This time the bottom lines of the two triangles are parallel.
(17) Lee Well, we still don’t know for sure whether that matters; that’s still just a conjecture. But we can check to see if the overlapping sides are proportional. [writes the following] $$\begin{align*} \frac{12.8}{8}=1.6 && \frac{8}{5}=1.6 \end{align*}$$ It works! They’re proportional.
(18) Matei So… side $BC$ corresponds with side $EC$, and side $AC$ corresponds with side $DC$.
(19) Lee And the scale factor is $1.6$.
(20) Chris So, what’s the length of side $CD$?

The teacher reflection questions are intended to prompt thinking about 1) the mathematical practices, 2) the mathematical content and extensions, 3) student thinking, and 4) teaching practices. Begin by reflecting on each of the questions, referring to the student dialogue as needed. After you had some time to do your own reflection, read the possible responses and comment at the bottom of the page with questions or thoughts provoked by the reflection questions and/or responses. Please note some of the mathematics extension tasks are meant for teacher exploration, to prompt engagement in the mathematical practices, and may not be appropriate for student use.

Teacher Reflection Questions

  1. What evidence do you see of students engaging in the Standards for Mathematical Practice?

    Possible Response

  2. Initially, students try folding to see whether a reflection would superimpose corresponding sides. They decide—first by appearance and then by checking scale factors—that reflection does not work in this case, but that rotation will. Is there a case in which reflection would superimpose corresponding sides, but rotation would not? If so, give an example; if not, show why not.

    Possible Response

  3. What challenges and/or misconceptions might students have when working with similar triangles?

    Possible Response

  4. The dialogue ends with Chris reminding the group of the original, still unanswered question. What MP is Chris meeting here? If students in your class had reasoned their way through the problem as the students in the dialogue do and yet were unable to answer the question that Chris recalls, how might you help?

    Possible Response

  5. How do the tools the students use allow them to continue reasoning about the problem or to engage in MPs? Are there additional tools you would recommend for students when working on this task?

    Possible Response

  6. In lines 11 and 16, Matei claims that if the sides that “line up” are corresponding sides, then the third sides must be parallel. In line 12, Chris questions whether the converse is true: If the third sides are parallel, does that guarantee that the superimposed sides correspond?

    Possible Response

  7. The students in the dialogue calculate the scale factors and test for proportionality, determining arithmetically which sides correspond. What information could be given in the task to let students determine geometrically (that is, without calculating the scale factors) which sides are corresponding?

    Possible Response

  8. How might you modify this mathematics task to make it more accessible or more challenging and thus suit a wide variety of students?

    Possible Response

The mathematical overview is intended to 1) explain how students in the dialogue engaged in the mathematical practices and 2) further explore the mathematical content and its extensions. Read the mathematical overview and comment at the bottom of this page with questions or thoughts it provoked.

Mathematical Overview

Mathematical Practice Evidence

Make sense of problems and persevere in solving them.

In lines 1–4 of the dialogue, we see students making sense of the geometric information they have been given (that the two triangles are similar) and the implication of that information (that corresponding sides will be proportional). Another important component of MP 1 is persevering in solving problems. In this dialogue, students try different transformations of $\triangle{ABC}$ to find the corresponding sides. After each transformation, they check to see if the sides that seem to “line up” are in fact corresponding (i.e., proportional). We also see students conjecturing about the possible meaning of having a set of parallel sides in the transformed nested triangles (lines 11–12).

Use appropriate tools strategically.

Using tracing paper to test their ideas is a strategic choice. Tracing paper is a simple low-tech tool to help organize students’ search and help them test out both reflections and rotations. They also use it to show each other what they mean, adding precision to their communication as they try to articulate ideas about which sides should be compared, and aiding understanding for all three students.

Attend to precision.

Imagine how this dialogue might have been different if the students were using a less precise notion of similarity (e.g., “same shape, different size”). The students are able to continue refining their ideas because they have some notions and questions (and share those notions and questions with each other) about the more precise meaning of similarity—they are examining correspondence through geometric transformations, and they are developing their understanding of how similarity involves scaling or dilation by a scale factor. The students use a mixture of informal and formal language as they communicate with each other about their ideas—talking about scaling, flipping, reflecting, and rotating. The students demonstrate attention to precision when they push each other to explain further or demonstrate (with appropriate tools! See MP 5) their understanding of these terms and to articulate more clearly their ideas. For example, in casual speech, you can’t be sure whether “turn the letter R upside down” means or , and the same is true of “flip the letter R upside down.” When Chris talks about “flipping” in line 5, Matei questions Chris about what it means to flip $\triangle{ABC}$ and Lee uses the tracing paper to check that they have the same definition of “flipping.” Chris then clarifies with more precise language in line 8, “Yeah, that’s what I meant. You reflected the triangle.”

Look for and make use of structure.

Students are using structure in their implicit understanding that a series of transformations can be used to change $\triangle{ABC}$ into $\triangle{DEC}$. Even though students do not perform the final dilation, searching for the appropriate series of transformations is what they are actually doing when looking for the corresponding sides of the similar triangles. Also students are looking for structure when they conjecture about the implications of a parallel vs. nonparallel third set of sides in the nested triangles (lines 11–12, 16).

Commentary on the Mathematics

Evidence of the Content Standards – Understanding similarity as a series of geometric transformations (8.G.A.4, G-SRT.A.2, and G-SRT.B.5)

Understanding congruence in terms of geometric transformations (i.e., that congruent figures are ones that can be superimposed on each other through a set of reflections, translations, and/or rotations—or “rigid” transformations) is a first step toward understanding geometric similarity. The next step is to consider the role of dilation: Two figures are similar if there exists a dilation of one that can be established as congruent to the other (again, by rigid transformations). In this problem, the students use the rigid transformations to help them determine which sides are corresponding so they can be compared through dilation. While the students in this dialogue do not explicitly talk about dilation, they are using the properties of dilation when they refer to scaling or the proportionality of figures to test corresponding sides. Just as rotations about a point and translations with respect to a point generate infinitely many images of an object, infinitely many similar figures (or dilated images) can be produced through dilation about a given point, from tiny to large. All along the dotted lines below—rays originating at the center of dilation (point $P$) and passing through the vertices of the shaded triangle—are the vertices of triangles similar to the shaded one. Conceptualizing infinite sets like this is the basis of a kind of abstract reasoning required in geometry. Of course, vertices of non-similar triangles can also be set on those rays (try it!), but their distances from the center of dilation then don’t fit the proportional pattern that characterize the vertices of all the similar triangles.

Shrinking down to a point—an image of change as a continuous process—can be extended to show a triangle “coming out the other side” of that point (extending the rays into lines). That is also a dilation by a negative scale factor. If the original triangle “comes out the other side” just far enough to achieve its original size, it appears to have been rotated $180$ degrees. Relationships among the transformations are interesting to explore—and not all are continuous. Reflection on the plane, for example, has a discrete nature: While one can rotate more or less, or dilate more or less, reflection in the plane is all or nothing.

Student discussion questions and related mathematics tasks are supplementary materials intended for optional classroom use with students. If you choose to use this task and dialogue with your students these discussion questions are intended to stimulate discussion and further exploration of the mathematics. Related mathematics tasks are intended to provide students an opportunity to engage in the mathematical practices as they connect to content that is similar to or an extension of that found in the dialogue. Please note, responses are to be read by teachers only in preparation for using the questions and tasks with their class.

Student Materials

Student Discussion Questions

  1. The dialogue ends with Chris asking, “What’s the length of side $CD$?” What is it? Explain your reasoning.

    Possible Response

  2. How do students use reflections and rotations in this dialogue? What other transformation would students need to perform at the end of the dialogue to turn $\triangle{ABC}$ into $\triangle{DEC}$?

    Possible Response

  3. In lines 11 and 16, Matei claims that if the sides that “line up” are corresponding sides, then the third sides must be parallel. In line 12, Chris questions whether the converse is true: If the third sides are parallel, does that guarantee that the superimposed sides correspond? Does it?

    Possible Response

  4. If you start with a square, can you create a similar square by increasing the length of each side by the same amount (e.g., adding two inches to each side)? What if you start with a non-square rectangle? Will increasing the length of each side by the same amount result in a similar rectangle? In each case, explain why or why not.

    Possible Response

Related Mathematics Tasks

  1. Figures $A$ and $B$ are similar. What is the length of $\overline{HG}$? Explain your reasoning.

    Possible Response

  2. How can you check if two circles are similar? How can you check if two circles are congruent?

    Possible Response

  3. Two similar triangles have areas of $4$ square inches and $36$ square inches. If the length of one side of the smaller triangle is $1.5$ inches, what is the length of the corresponding side of the larger triangle?

    Possible Response

  4. Jose is wondering how far apart two docks are on the other side of a river. He knows that the river is $300$ yards across at all points in this section of the river. He has a measuring tape so he can measure distances on his side of the river. How can he use similar triangles to find out the distance between the two docks?

    Possible Response

  • MP 1: Make sense of problems and persevere in solving them.

    MP 1: Make sense of problems and persevere in solving them.

    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

  • MP 5: Use appropriate tools strategically.

    MP 5: Use appropriate tools strategically.

    Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

  • MP 6: Attend to precision.

    MP 6: Attend to precision.

    Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

  • MP 7: Look for and make use of structure.

    MP 7: Look for and make use of structure.

    Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(xy)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.