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Unit 4, Lesson 5

Working with Ratios in Right Triangles

Geometry

5.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Scientific calculators

Activity Narrative

In a previous lesson’s Cool-down, students studied a similar situation involving a jet airplane rather than a rocket. In that Cool-down students were given a scale diagram of the situation. Students may have used estimation strategies or they may have calculated ratios of side lengths and used the Right Triangle Table to estimate a range of possible values for the launch angle. Throughout this Warm-up, encourage students to think more about how calculating ratios and looking at the Right Triangle Table can help them make a more accurate estimate.

Launch

Ask students if they have watched a rocket launch or heard about mathematicians, such as Katherine Johnson, who do the calculations required for launching rockets. Large rockets from NASA are usually launched vertically and change angles in flight. Model rockets are typically launched at an angle such as the one in this situation.

Ensure that students have access to their Right Triangle Table for this activity and the remainder of the lesson.

Activity

None

When a rocket is launched, it climbs 50 feet for every 13 feet it travels horizontally. Draw a diagram to represent the situation. Then estimate the rocket’s launch angle.

Student Response

to access Student Response.

Building on Student Thinking

If students are struggling to make reasonable estimates for the angle measures, refer them to their Right Triangle Table.

Activity Synthesis

The purpose of this discussion is to review strategies for estimating angle measures and to remind students that the Right Triangle Table is a useful tool for this.

Invite a student to share their method. (I used the Right Triangle Table and compared the ratio of the sides to the ratios in the table.) Remind students to use their Right Triangle Table as reference throughout class.

5.2

Activity

Instructional Routines

None

Materials

To Gather

Scientific calculators

Activity Narrative

In this activity, students estimate angle measures in right triangles by calculating the ratios of side lengths and using their Right Triangle Table to associate those ratios with angle measures.

Monitor for students who use one of these strategies to share during the Activity Synthesis:

Calculate the ratios of side lengths and compare them to ratios in the Right Triangle Table.
Use complementary angles to find the second angle measure once the first has been established.

Launch

Ask students what the Pythagorean Theorem says. (If a right triangle has legs and and hypotenuse , then .) Ask students what the converse of the Pythagorean Theorem says. (If a triangle has sides , and that satisfy , then it is a right triangle.)

Tell students, “That means a triangle with side lengths 7, 24, and 25 is a right triangle because . A set of whole numbers that satisfies the converse of the Pythagorean Theorem is called a Pythagorean triple.”

5.3

Activity

Instructional Routines

MLR7: Compare and Connect Notice and Wonder

Materials

To Gather

Scientific calculators

Activity Narrative

This activity is the first time students are expected to use trigonometric ratios rather than a specific similar triangle to find unknown side lengths in right triangles. An important aspect of the activity is to highlight strategies students use since different strategies will make sense to different students depending on how they make connections between the ratios and the right triangles they represent (MP7).

Monitor for students who:

Use the information from the Right Triangle Table as constants of proportionality and work with relationships.
Use the information from the Right Triangle Table as ratios and find a scale factor that scales them up to the size of the given triangle.
Use the information from the Right Triangle Table as values, and guess and check to find a side length that results in the right value.

Students need only one strategy, but it’s important for each student to have at least one strategy they can make sense of.

This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.

Lesson Synthesis

Remind students of all the awesome things they can calculate using the Right Triangle Table they built themselves. They can find the launch angles for jet airplanes and rockets. They can find unknown side lengths in right triangles with only one side and one angle given, without having another similar triangle to compare it to.

Invite students to respond in writing to the prompts:

Why does having the Right Triangle Table give you enough information to find unknown lengths and angle measures in right triangles with only two other measurements given?
In what ways is the Right Triangle Table helpful?
What is hard about using the Right Triangle Table right now?

to access Cool-down.

Student Lesson Summary

All right triangles that contain the same acute angle are similar. This means that the ratios of corresponding side lengths are equal for all right triangles with the same acute angles. Using the ratios calculated in the previous lesson and properties of similar triangles, we can calculate and estimate unknown side lengths and angles in right triangles.

If we measure the legs of any right triangle with an angle of 25 degrees, the ratio of the leg opposite the 25-degree angle to the leg adjacent to the 25-degree angle will always be 0.466. Therefore, we can find length . Since , we know is 10.7 units.

<p>Right triangle ABC. Side BC = 5 units. Angle A = 25 degrees. </p>

Similarly, we can estimate the measure of the unknown angles in triangle .

angle	adjacent leg hypotenuse	opposite leg hypotenuse	opposite leg adjacent leg
	0.643	0.766	1.192
	0.500	0.866	1.732

The ratio of the leg opposite angle to the hypotenuse is 0.794. This value is between the value of the ratio of the opposite leg divided by hypotenuse for 50 degrees (0.766) and 60 degrees (0.866). So the measure of angle must be between 50 and 60 degrees. Similarly, the leg opposite angle divided by the leg adjacent to angle gives a ratio of 1.283, which is between the same ratio for 50 degrees (1.192) and 60 degrees (1.732). The exact value turns out to be 52 degrees.

<p>Right triangle DEF. DE= 6 units, EF = 7 point 7 units, FD = 9 point 7 units. Right angle E.</p>

Launch

Tell students to close their books or devices (or to keep them closed).

Display this image for all to see:

Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image.

Things students may notice:

It’s a right triangle.
One angle is 20 degrees.
The other acute angle is 70 degrees.
One side is 8.5 units long.
Two of the sides are unknown.

Things students may wonder:

Are they going to ask us to find the lengths of the unknown sides?
Is this triangle on the table?
Do we need more information?

If using the Right Triangle Table to find a unknown side length does not come up during the conversation, ask students to discuss this idea.

Select students who used each strategy described in the Activity Narrative to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially of students who haven't shared recently.

Action and Expression: Internalize Executive Functions. To support organization, provide students with a four-column table. Use these column headings: “angle,” “adjacent leg,” “opposite leg,” and “hypotenuse.” The table will provide visual support for students to identify ratios.
Supports accessibility for: Language, Organization

Activity

None

What is the length of segment ?
In a right triangle with one angle measuring 40 degrees, the leg opposite the 40-degree angle is 5 cm. What is the length of the hypotenuse?
What is the length of segment ?
In a right triangle with one angle measuring 70 degrees, the leg opposite the 70-degree angle is 12 cm. What is the length of the leg adjacent to the 70-degree angle?

Student Response

to access Student Response.

Building on Student Thinking

Encourage students to draw a diagram to solve the problem if one is not provided.

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Lesson 5

Working with Ratios in Right Triangles

Warm-up

Standards Alignment

Building On

Addressing

HSG-SRT.C.6

Building Toward

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

8.G.B.7

Addressing

HSG-SRT.C.6

Building Toward

Standards Alignment

Building On

Addressing

HSG-SRT.C.6

Building Toward