Lesson 8
Finding Unknown Side Lengths
8.1: Which One Doesn’t Belong: Equations (5 minutes)
Warmup
The purpose of this warmup is to prime students for solving equations that arise while using the Pythagorean Theorem.
Launch
Arrange students in groups of 2–4. Display the equations for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reason why a particular equation does not belong and together find at least one reason each question doesn’t belong.
Student Facing
Which one doesn’t belong?
\(3^2 + b^2 = 5^2\)
\(b^2 = 5^2  3^2 \)
\(3^2 + 5^2 = b^2\)
Student Response
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Activity Synthesis
Ask each group to share one reason why a particular equation does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given make sense.
8.2: Which One Is the Hypotenuse? (5 minutes)
Activity
This activity helps students identify the hypotenuse in right triangles in different orientations.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet work time and then have them compare with a partner. Follow with a wholeclass discussion.
Student Facing
Label all the hypotenuses with \(c\).
Student Response
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Activity Synthesis
Ask students which triangles are right triangles, and then ask them which side is the hypotenuse for each one. Ask, “In a right triangle, does it matter which is \(a\) and which is \(b\)?” (No.)
Design Principle(s): Support sensemaking
8.3: Find the Missing Side Lengths (20 minutes)
Activity
The purpose of this activity is to give students practice finding missing side lengths in a right triangle using the Pythagorean Theorem.
Launch
Arrange students in groups of 2. Give students 10 minutes of quiet work time and then have them compare with a partner. If partners disagree about any of their answers, ask them to explain their reasoning to one another until they reach agreement. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Student Facing
 Find \(c\).
 Find \(b\).
 A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?
 A right triangle has a side of length \(\frac14\) and a hypotenuse of length \(\frac13\). What is the length of the other side?

Find the value of \(x\) in the figure.
Student Response
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Student Facing
Are you ready for more?
The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.
Find the length, \(x\), of the hypotenuse of the last triangle constructed in the figure.
Student Response
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Activity Synthesis
Ask students to share how they found the missing side lengths. If students drew triangles for the two questions that did not have an image, display a few of these for all to see, noting any differences between them. For example, students may have drawn triangles with different orientations or labeled different sides as \(a\) and \(b\).
For the last question, ask students to say what they did first to try and solve for \(x\). For example, while many students may have found the length of the unknown altitude first and then used that value to find \(x\), others may have set up the equation \(345^2=18x^2\).
Point out that when you know two sides of a right triangle, you can always find the third by using the Pythagorean identity \(a^2 + b^2 = c^2\). Remind them that it is important to keep track of which side is the hypotenuse.
Design Principles(s): Cultivate conversation; Maximize metaawareness
Lesson Synthesis
Lesson Synthesis
The purpose of this discussion is to check that students understand the Pythagorean Theorem and how it can be used to determine information about triangles. Ask students to draw a right triangle and label 2 of the 3 sides. Tell them to swap triangles with another student, solve for the missing length, then swap back to check the other person’s work. Select a few groups to share their triangles and, if possible, display them for all to see while sharing how they solved for the unknown length.
8.4: Cooldown  Could be the Hypotenuse, Could be a Leg (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
There are many examples where the lengths of two legs of a right triangle are known and can be used to find the length of the hypotenuse with the Pythagorean Theorem. The Pythagorean Theorem can also be used if the length of the hypotenuse and one leg is known, and we want to find the length of the other leg. Here is a right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by \(g\).
Start with \(a^2+b^2=c^2\), make substitutions, and solve for the unknown value. Remember that \(c\) represents the hypotenuse: the side opposite the right angle. For this triangle, the hypotenuse is 10.
\(\begin{align} a^2+b^2&=c^2 \\ 5^2+g^2&=10^2 \\ g^2&=10^25^2 \\ g^2&=10025 \\ g^2&=75 \\ g&=\sqrt{75} \\ \end{align}\)
Use estimation strategies to know that the length of the other leg is between 8 and 9 units, since 75 is between 64 and 81. A calculator with a square root function gives \(\sqrt{75} \approx 8.66\).