Lesson 5

Working with Ratios in Right Triangles

5.1: Launch Pad (5 minutes)

Warm-up

In the previous lesson’s cool-down, students studied a similar situation involving a jet airplane rather than a rocket. In the 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. Through this warm-up and the synthesis, 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

Ensure that students have access to their right triangle table for this activity and the remainder of the lesson.

Student Facing

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

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are struggling to make reasonable estimates for the angle measures, refer them to their
right triangle table.

Activity Synthesis

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

5.2: Pythagorean Triples (15 minutes)

Activity

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. In previous activities, students were estimating angle measures in a context. In this activity, they are estimating angle measures in right triangles with whole-number side lengths.

Monitor for:

  • students calculating the ratios of side lengths and comparing them to ratios in the right triangle table
  • students using 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 \(a\) and \(b\) and hypotenuse \(c\), then \(a^2+b^2=c^2\).) Ask students what the converse of the Pythagorean Theorem says. (If a triangle has sides \(a\)\(b\) and \(c\) that satisfy \(a^2+b^2=c^2\), then it is a right triangle.)

Tell students, “That means a triangle with side lengths 7, 24, and 25 is a right triangle because \(7^2+24^2 =25^2\). A set of whole numbers that satisfies the converse of the Pythagorean Theorem is called a Pythagorean triple.”

Student Facing

  1. Sketch the triangle with side lengths 7, 24, and 25 units. Label the smallest angle \(A\).
  2. Find the 3 ratios of side lengths for angle \(A\).
  3. Estimate the acute angles in this triangle.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Find another Pythagorean triple.
  2. Estimate the acute angles in that triangle. Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are struggling to make reasonable estimates for the angle measures, refer them to their right triangle table. 

Activity Synthesis

Invite students who used the right triangle table to find the angle measures by comparing and ordering the ratios of side lengths to explain their reasoning. 

Ask students if they repeated the process for the other acute angle. (No, once I know one acute angle in a right triangle the other one has to be complementary. Yes, if I check those ratios too I might be able to make a better estimate.)

5.3: Solve All the Triangles (15 minutes)

Activity

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.

Monitor for students who:

  • use the information from the right triangle table as constants of proportionality and work with \(y=kx\) 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

Select students to share each strategy, as different strategies will make sense to different students depending on how they conceptualize the relationships. Students only need one strategy, but it’s important for each student to have at least one strategy they can make sense of.

Launch

Display this image for all to see:

A right triangle.

Ask students to think of at least one thing they notice and at least one thing they wonder.

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 missing sides?
  • Is this triangle on the table?
  • Do we need more information?

If using the right triangle table to find a missing side length does not come up during the conversation, ask students to discuss this idea.

Action and  Expression: Internalize Executive Functions. Provide students with a four-column table to organize. Use the column headings: angle, adjacent side, opposite side and hypotenuse. The table will provide visual support for students to identify ratios.
Supports accessibility for: Language; Organization

Student Facing

  1. What is the length of segment \(AB\)?

    A right triangle.
  2. 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?
  3. What is the length of segment \(DE\)?

    Right triangle DEF. side EF = 6. Angle F = 50 degrees. 
  4. 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

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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

Activity Synthesis

Invite students to share their strategies. If a strategy is not used by any student it is not necessary to mention it.

Possible strategies:

  • using the information from the right triangle table as constants of proportionality and working with \(y=kx\) relationships
  • using the information from the right triangle table as ratios and finding a scale factor that scales them up to the size of the given triangle
  • using the information from the right triangle table as values and guessing and checking to find a side length that results in the right value
Speaking, Listening: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each strategy that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

Lesson Synthesis

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 drawn 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 2 other measurements given?
  • What are you good at using the right triangle table to figure out?
  • What is hard about using the right triangle table right now?

5.4: Cool-down - Solve the Triangle (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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 \(b\). Since \(\frac{5}{b} = 0.466\), we know \(b\) is 10.7 units.

Right triangle ABC. Side BC = 5 units. Angle A = 25 degrees. 

Similarly, we can estimate the measure of the missing angles in triangle \(DEF\).

angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
\(50^\circ\) 0.643 0.766 1.192
\(60^\circ\) 0.500 0.866 1.732

The ratio of the leg opposite angle \(D\) to the hypotenuse is 0.794. This value is between the value of opposite leg divided by hypotenuse for 50 degrees (0.766) and 60 degrees (0.866). So the measure of angle \(D\) must be between 50 and 60 degrees. Similarly, the leg opposite angle \(D\) divided by the leg adjacent to angle \(D\) 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.

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