Unit 2, Lesson 17
Finding All the Unknown Values in Triangles
Integrated Math 2
17.1
Warm-up
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
NoneActivity
NoneWhich three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any geometric terminology they use, such as “right triangle” or “horizontal,” and to clarify their reasoning as needed. Consider asking:
- “How do you know_____?”
- “What do you mean by _____?”
- “Can you say that in another way?”
Lesson Synthesis
“When finding all the unknown values in similar right triangles, you have choices: You can use proportional relationships or the Pythagorean Theorem. Which would you use for each of these examples, and why?”
(The larger triangle is a scaled copy of the smaller one with a scale factor of 2, so it’s straightforward to find the missing lengths by multiplying the corresponding lengths by 2, so is 6 and is 8.)
( is a right triangle so I can use the Pythagorean Theorem to find the missing length. That’s more direct than finding the scale factor between 8.45 and 13. is 12. Finding will involve fractions either way, but I’ll use the scale factor from 13 to 8.45 and apply that to and . is 3.25.)
After students choose and defend their thinking, encourage them to check each other’s work using another method. Why should both methods give the same answer? (The side lengths don’t change just because the way that we calculate them changes.)
Student Lesson Summary
We have multiple strategies to find unknown side lengths in similar right triangles.
Two right triangles, F G H and A B C. Angles G F H and B A C are marked congruent, Angles F G H and A B C are marked right angles. Side F G labeled y, Side G H labeled 12 over 5, Side F H labeled 4. Side A B labeled 4, side B C labeled 3, Side C A labeled 5.
Because triangle is a right triangle, . Because triangle is a right triangle, . Because triangle is similar to triangle , there are many equations to write using proportional relationships. We can use any combination of these equations to solve for and .
By similarity, . Substituting into the Pythagorean Theorem gives which is true.
By the Pythagorean Theorem, , so . By similarity , which also equals 3.