This Warm-up prepares students for the argument of the converse of the Pythagorean Theorem that will be constructed in the next activity. The Warm-up relies on the Pythagorean Theorem and geometrically intuitive facts about how close or far apart the two hands of a clock can get from one another.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time, and follow with a whole-class discussion.
Activity
None
Student Lesson in Spanish
Consider the tips of the hands of an analog clock that has an hour hand that is 3 centimeters long and a minute hand that is 4 centimeters long.
The image of a circle that represent an analog clock. On the circle are 12 evenly spaced tick marks. There are two hands on the clock. One hand is labeled 3, begins in the center of the circle and extends upward and to the right, and points to the third tick mark from the top. The other hand is labeled 4, begins in the center of the circle and extends upward and to the left. It points to the eleventh tick mark from the top.
Over the course of a day:
What is the farthest distance apart the two tips get?
What is the closest distance the two tips get?
Are the two tips ever exactly five centimeters apart? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The focus of this discussion is to prepare students to follow a specific line of reasoning that is needed to make sense of the converse of the Pythagorean Theorem. First, invite 1–2 students to briefly share their answers to the three questions. Then ask students to consider the following line of reasoning:
Imagine two hands starting together, where one hand stays put and the other hand rotates around the face of the clock. As one hand rotates, the distance between its tip and the tip of the stationary hand continually increases until they are pointing in opposite directions. So from one moment to the next, the tips get farther apart.
(Proving this requires mathematics beyond grade 8, so for now we will just accept the results of the thought experiment as true.)
10.2
Activity
Standards Alignment
Building On
Addressing
8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
This activity introduces students to the converse of the Pythagorean Theorem: In a triangle with side lengths , , and , if we have , then the triangle must be a right triangle, and must be its hypotenuse.
This may be the first time students have considered the idea that a theorem might work one way but not the other. For example, it is not clear at first glance that there is no such thing as an obtuse triangle with side lengths 3, 4, and 5, as in the image. But since , the converse of the Pythagorean Theorem will say that the triangle must be a right triangle.
The argument this activity presents is based on the thought experiment introduced in the Warm-up. As the angle created by two sides of length 3 and 4 increases, the distance between their endpoints also increases, from a distance of 1 when they are pointing in the same direction to a distance of 7 when pointing in opposite directions. There is then only one angle along the way where the distance between them is 5, and by the Pythagorean Theorem, this happens when the angle between them is a right angle. This argument generalizes to any arbitrary right triangle, proving that the only angle that gives is precisely the right angle.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet work time, and follow with a whole-class discussion.
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, use GeoGebra or hands-on manipulatives to demonstrate how increasing or decreasing the angle between 2 sides affects the opposite side length. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Activity
None
Here are three triangles with two side lengths measuring 3 and 4 units, and the third side of unknown length.
A figure of three triangles each with 2 given side lengths and one unknown side length. The first triangle has a horizontal side of 4, a side length slanted upward and to the left of 3, and the third side length labeled x. The middle triangle has a horizontal side length of 4, a second side length slanted upward and to the right of 3, and the third side length labeled y. The third triangle is a right triangle with a horizontal side length of 4, a vertical side length of 3, and the third side is labeled z.
Order the following six values from smallest to largest. Put an equal sign between any you know to be equal. Be prepared to explain your reasoning.
Activity Synthesis
The goal of this discussion is for students to understand that given a triangle with two side lengths of 3 and 4, there is only one angle between 0 and 180 degrees where the third side has a length of 5. Begin by inviting students to share their order of the 6 values. Consider discussing the following questions:
“Why is 1 the smallest value?” (The distance between the two vertices of a triangle with legs of length 3 and 4 could only be 1 if the two legs were on top of each other.)
“Why is 7 the largest value?” (The distance between the two vertices of a triangle with legs of length 3 and 4 could only be 7 if the two legs were facing opposite directions.)
“Why is ?” (Since is the hypotenuse of a right triangle, by the Pythagorean Theorem its length must be 5.)
Then display this image for all to see.
Make sure students understand the following sequence of ideas:
As we saw with the clock problem, the length of the third side continually increases from 1 to 7 as the angle increases between and .
Because of this, there is only one angle along the way that gives a third side length of 5.
By the Pythagorean Theorem, if the angle is a right angle, the third side length is 5.
Then display this image for all to see.
Discuss how the argument could be made with any two starting side lengths and by asking the following questions:
“What is the smallest possible distance between the tips of and ?” (When sides and are on top of each other, or when the angle between the two sides is , the distance would be or .)
What is the greatest possible distance between the tips of and ? (When the angle between and is , the distance between the tips is or .)
“Between and , how many times will the distance between the tips of and be the same?” (The distance between the tips will never be the same.)
“Between and , how many times will ?” (Only once, when the angle between and is .)
Conclude that the only for is if the triangle is a right triangle, with hypotenuse . This result is called "the converse of the Pythagorean Theorem."
The purpose of this activity is for students to apply the Pythagorean Theorem and its converse in mathematical contexts. In the first problem, students apply the Pythagorean Theorem to find unknown side lengths. In the second problem, students use the fact that by changing side lengths so that they satisfy , the resulting triangle is guaranteed to be a right triangle.
Monitor for students who solve the second problem by:
Using 4 and 6 as legs and solving for the hypotenuse.
Using 4 and 7 as legs and solving for the hypotenuse.
Using 6 and 7 as legs and solving for the hypotenuse.
Using 7 as the hypotenuse and either 4 or 6 as one of the legs and solving for the other leg.
Using 6 as the hypotenuse and 4 as the leg and solving for the other leg.
While it is not necessary to discuss all of these different approaches, monitor for at least one where the value changed is a leg and one where it is the hypotenuse.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Provide students with access to calculators.
Display the image from the first question. Tell students that the right angle is not labeled, but the prompt tells us these are both right triangles. Ask, “How do we know which side of a right triangle is the hypotenuse?” (It is the longest side of the right triangle.)
Give students 4–5 minutes of quiet work time, and follow with a whole-class discussion.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Activity
None
Given the information provided for the right triangles shown here, find the unknown leg lengths to the nearest tenth.
Two right triangles are indicated. The triangle on the left has two leg with lengths of 2 and a. The hypotenuse has a length of 7. The triangle on the right has two legs with length x and a hypotenuse of length 6.
The triangle shown here is not a right triangle. What are two different ways you change one of the values so it would be a right triangle? Sketch these new right triangles, and clearly label the right angle.
Student Response
Activity Synthesis
The goal of this discussion is for students to share how they used the Pythagorean Theorem to reason about the side lengths of right triangles. Select 1–2 students to share their work for the first problem.
For the second problem, display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. It is not necessary to go through all of the possibilities—the focus here is to see how the various equations of the form informed how they sketched their right triangles. Here are some questions for discussion:
“What do the approaches have in common? How are they different?”
“What kinds of additional details or language helped you understand the displays? Are there any additional details or language that you have questions about?”
“Did anyone solve the problem the same way, but would explain it differently?”
If time allows:
How many different ways are there to solve this problem? (6 ways)
Lesson Synthesis
The purpose of this discussion is to summarize the Pythagorean Theorem and its converse. Begin by displaying this image:
Discuss:
“Do you think this triangle is a right triangle or not? Explain your reasoning.” (Answers vary.)
“How can we tell for sure if this is a right triangle, even though no right angle is indicated?” (We can use the Pythagorean Theorem and see if the sum of the squares of the two shorter sides equals the square of the longest side.)
Give students 1–2 minutes of quiet work time to see if the Pythagorean Theorem holds for this triangle. Emphasize that while the angle across from the side of length 10 looks like a right angle, we can't be sure it is just from the image.
Invite 1–2 students to share their work showing that the triangle is a right triangle, or display the following for all to see:
Since is true, we know, by the converse of the Pythagorean Theorem, that the triangle is a right triangle and that the angle across from the side of length 10 is a right angle.
Student Lesson Summary
How can we tell whether a triangle is a right triangle or not? For example, in this triangle it isn’t clear just by looking, and it may be that the sides aren’t drawn to scale.
If we have a triangle with side lengths , , and , with being the longest of the three, then the converse of the Pythagorean Theorem tells us that any time we have , we must have a right triangle. Since , any triangle with side lengths 8, 15, and 17 must be a right triangle.
What about the jib sail on this boat? It is a triangle, but is it a right triangle?
The measurements of the jib sail are shown here. The sum of the squares of the two shorter sides is 106.965 square meters, and the square of the longest side is 104.8576 square meters. So by the converse of the Pythagorean Theorem, it is not a right triangle, but it is close to one.
Together, the Pythagorean Theorem and its converse provide a way to check if a triangle is a right triangle just using its side lengths. If , it is a right triangle. If , it is not a right triangle.
Standards Alignment
Building On
Addressing
Building Toward
8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
