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 and are two side lengths of a triangle, then the corresponding side lengths of a similar triangle have lengths and for some positive scale factor , and the ratios and are equivalent. By repeatedly dividing one side length of a triangle by another side length of the same triangle, students determine that the quotients of pairs of side lengths in similar triangles are equal (MP8).

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) (long side) be?” ()

• “Do you think the value of (medium side) (long side) would be for any triangle similar to triangle ?” (Yes. Any triangle similar to will have side lengths that are multiples of 4, 5, and 7. The medium side divided by the long side will always be a fraction equivalent to .)

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)

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 , , and all red.

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 has 2 equal side lengths, so the other 2 triangles will as well. This insight is efficient for finding . 

• One side of triangle is twice the length of another side, so this will be true for the other triangles as well. This insight is helpful for finding , , and .