Unit 8, Lesson 7
Finding Side Lengths of Triangles
Grade 8
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This Warm-up prompts students to carefully analyze and compare features of triangles. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of triangles.
Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three triangles that go together and can explain why. Next, tell students to share their response with their group, and then together find as many sets of three as they can.
Which three go together? Why do they go together?
First of four triangles is a right triangle, legs = 5, 5. hypotenuse = square root 50.
Second of four triangles has sides = 2, 5, square root 33.
Third of four triangles is a right triangle, legs = 1,8.
Fourth of four triangles is a right traingle, legs = 3,4. hypotenuse = 5.
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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. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as irrational, scalene, or isosceles, and to clarify their reasoning as needed. Consider asking:
MLR2: Collect and Display
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The purpose of this activity is for students to think about the relationships between the squares of the side lengths of triangles as a lead up to the Pythagorean Theorem at the end of this lesson.
Students record both the side lengths and areas of squares and look for patterns (MP8). Note that some side lengths are intentionally positioned so that students won’t be able to easily draw squares. In these cases, the segments are congruent to others whose lengths are already known or could be calculated. Some side lengths lie along gridlines.
Arrange students in groups of 2–3. Display the image of the triangle on a grid for all to see and ask students to consider how they would find the value of each of the side lengths of the triangle. Give students 1–2 minutes of quiet work time and then pause for a brief partner discussion before allowing groups to continue calculating the side lengths.
Select 2–3 groups to share their strategies and the values for the side lengths they found (
If not mentioned by students, note that for the side that lies along a gridline, the square is not needed to determine the side length. But since
Encourage groups to divide up the work completing the tables and to discuss strategies for finding the rest of the unknown side lengths.
The purpose of this discussion is to allow students to communicate what they noticed using precise mathematical language. Students may notice that
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The goal of this activity is to reach a formal statement of the Pythagorean Theorem. Students will work with a proof of the theorem in a later lesson, so the focus here is on building a basic understanding of what the theorem says and that it is not true for all triangles.
Arrange students in groups of 2. Give students 4 minutes of quiet work time followed by a partner discussion. Then follow with a whole-class discussion.
Invite selected students to share their strategies for determining the missing side lengths. Make sure the class comes to an agreement for which triangles
If
A right triangle is a triangle with a right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and the two other sides that make the right angle are called its legs.
Here are some right triangles with the hypotenuse and legs labeled:
If the triangle is a right triangle, then
Here are some other right triangles:
Three right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 25” is attached to the hypotenuse. The triangle in the middle has the square labels “a squared equals 16” and “b squared equals 1” attached to each of the legs. The square labeled “c squared equals 17” is attached to the hypotenuse. The triangle on the right has the square labels “a squared equals 9” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 18” is attached to the hypotenuse.
Notice that for these examples of right triangles, the square of the hypotenuse is equal to the sum of the squares of the legs. In the first right triangle in the diagram,
This is a property of all right triangles, not just these examples, and is often known as the Pythagorean Theorem. The name comes from a mathematician named Pythagoras who lived in ancient Greece around 2,500 BCE, but this property of right triangles was also discovered independently by mathematicians in other ancient cultures including Babylon, India, and China. In China, a name for the same relationship is the Shang Gao Theorem.
It is important to note that this relationship does not hold for all triangles. Here are some triangles that are not right triangles. Notice that the lengths of their sides do not have the special relationship
Two right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” aligned to the bottom horizontal leg and “b squared equals 10” aligned to the left leg. The square labeled “c squared equals 18 is aligned with the hypotenuse. The triangle on the right has the square labels of “a squared equals 10” aligned with the bottom leg and “b squared equals 2” aligned with the left leg. The square labeled “c squared equals 16” is aligned with the hypotenuse.
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Complete the tables for these three triangles:
Triangle on a square grid with sides a, b, and c. Horizontal side a is 2 units. Side b slants upward and to the left. Side c slants downward and to the right. The height of the triangle is 1.
Triangle on a square grid with sides a, b, and c. Horizontal side a is 2 units. Vertical side b is 1 unit. Diagonal side c slants downward and to the right and the triangle has a height of 1 unit.
Triangle on a square grid with sides a, b, and c. Horizontal side a is 2 units. Side b and side c are equal in length. The triangle has a height of 2 units.
| Triangle D | |||
|---|---|---|---|
| Triangle E | |||
| Triangle F |
| Triangle D | |||
|---|---|---|---|
| Triangle E | |||
| Triangle F |
Complete the tables for three more triangles:
Triangle on a square grid with sides a, b, and c. Horizontal side a is 2 units. Vertical side b is 3 units. Diagonal side c slants downward and to the right and the triangle has a height of 3 units.
Triangle on a square grid with sides a, b, and c. Horizontal side a is 2 units. Side b and side c are equal in length. The triangle has a height of 3 units.
| |
|
|
|
|---|---|---|---|
| Triangle P | |||
| Triangle Q | |||
| Triangle R |
| Triangle P | |||
|---|---|---|---|
| Triangle Q | |||
| Triangle R |
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