Lesson 4

Ratios in Right Triangles

4.1: Ratio Rivalry (5 minutes)

Warm-up

In order to make sense of trigonometry, students must be certain that right triangles with the same angle measures are similar, and therefore that ratios of side lengths in right triangles with the same angle measure, regardless of the size of the triangle, are equal. This warm-up challenges students to verify that ratios of side lengths in right triangles with the same angles are equal. Students are given the measure of only one of the acute angles, so they must rely on the Angle-Angle Triangle Similarity Theorem.

Student Facing

2 similar right triangles. Base angles = 20 degrees. Triangle on left, height = a, hypotenuse = c. Triangle on right, height = b, hypotenuse =d. 

Consider \(\frac{a}{c} \text{ and } \frac{b}{d}\). Which is greater, or are they equal? Explain how you know.

Student Response

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

Activity Synthesis

Ask students to convince someone who thought \(\frac{b}{d}\) was greater that in fact the two ratios are equal. Invite several students to share different ways of arguing that the fractions are equal, such as:

  • measure the lengths of \(a, b, c, \text{ and } d\) and compare the fractions
  • the triangles are similar by the Angle-Angle Triangle Similarity Theorem, so \(\frac{a}{b} = \frac{c}{d}\) and that means that \(\frac{a}{c} = \frac{b}{d}\) too
  • the triangles are similar by the Angle-Angle Triangle Similarity Theorem, so \(b=ka \text{ and } d=kc\) for some scale factor \(k\), so \(\frac{a}{c} = \frac{ka}{kc} = \frac{b}{d}\).
  • the triangles are similar by the Angle-Angle Triangle Similarity Theorem, so side lengths are proportional within and between triangles, so \(\frac{a}{c} = \frac{b}{d}\)

4.2: Tons of Triangles (15 minutes)

Activity

In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, by angle measure of one acute angle). For now, call this the right triangle table so students remember these ratios only apply to right triangles. Building these ratios from the similar triangles they are based on helps students build connections between similar triangles and trigonometry.

Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, it provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles with their angle as one of the angles will be similar and thus have the same ratios.

In the next activity and the cool-down of this lesson, students will look for patterns in the right triangle table, and also apply the ratios in the right triangle table to find unknown angle measures given a ratio of side lengths. Do not use the word trigonometry or any of the names of the ratios yet.

Launch

Provide access to devices that can run GeoGebra. Students will be working with the applet at ggbm.at/bfxeb6cr.

Arrange students in groups of 2-4. Invite students to create a spreadsheet to organize their data and perform calculations. Students can access a GeoGebra spreadsheet in the math tool kit or at geogebra.org/spreadsheet. For students needing extra support, consider providing this spreadsheet for them to use: ggbm.at/jvvqrngb.

Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.

Explain to students that the computer measurement is precise to at least the ten-thousandths place, but the thousandths place is sufficient for our needs.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle \(A\), the side opposite angle \(A\), and the hypotenuse. 
Supports accessibility for: Visual-spatial processing.

Student Facing

Your teacher will give you some angles.

  1. Use the applet to build 4 different right triangles for each of your angles.
  2. Record the side lengths of each of the triangles.
  3. Compute these 3 quotients for the acute angles in each triangle:
    1. The length of the leg adjacent to your angle divided by the length of the hypotenuse
    2. The length of the leg opposite from your angle divided by the length of the hypotenuse
    3. The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
  4. Find the mean of each type of quotient.
  5. What do you notice? What do you wonder?

Student Response

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

Launch

Arrange students in groups of 3-4. Give each group a copy of the blackline master to complete.

Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.

Discuss rounding with students before they begin. Ask students whether it makes sense to measure the short side of a triangle as about 5.2 cm and the hypotenuse as pretty close to 7.1 cm and then report that the ratio of the short side to the hypotenuse is exactly 0.73239437. (No, that's too many digits to report.) 

Explain to students that the rule of thumb used by scientists is that if your measurements have two digits then your calculated answer should have two digits as well. For example, if your tools measure to the tenths place, say 5.2 cm, so that your measurements have one digit you’re confident in - the 5 in the one’s place, and one digit you may have had to estimate a bit - the 2 in the tenths place, then your answer should be rounded to two digits - 0.73. In the case where the number starts with a zero we count digits starting at the first non-zero digit, so in this case we will round to the hundredths place.

As students finish their measurements, have them find the mean of each column and record that on the class Right Triangle Table.

Check students’ mean data before they enter it to ensure that it is close to the actual values:

angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
10 degrees 0.985 0.174 0.176
20 degrees 0.940 0.342 0.364
30 degrees 0.866 0.500 0.577
40 degrees 0.766 0.643 0.839
50 degrees 0.643 0.766 1.192
60 degrees 0.500 0.866 1.732
70 degrees 0.342 0.940 2.747
80 degrees 0.174 0.985 5.671

This table is needed for the next activity, so it should be recorded to display at that time.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle \(A\), the side opposite angle \(A\), and the hypotenuse. 
Supports accessibility for: Visual-spatial processing.

Student Facing

Your teacher will give you some angles.

  1. Draw several right triangles using the angles you receive.
  2. Precisely measure the side lengths of the triangles.
  3. Complete the tables by computing 3 quotients for the acute angles in each triangle:
    1. The length of the leg adjacent to your angle divided by the length of the hypotenuse
    2. The length of the leg opposite from your angle divided by the length of the hypotenuse
    3. The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
  4. Find the mean of each column in your table.
  5. What do you notice about your table? What do you wonder about your table?

Student Response

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

Anticipated Misconceptions

Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” then the leg you are touching is adjacent and the leg you are not touching is opposite. The hypotenuse is always the longest side.

A right triangle, annotated.

Activity Synthesis

Invite groups to share what they noticed and wondered about their data. (Each ratio is about the same for the angle. The ratio of the opposite side to the hypotenuse for angle \(A\) is the same as the ratio of the adjacent side to the hypotenuse for angle \(B\).)

Speaking: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain what they noticed and wondered about the relationships between the quantities represented in the table. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation

4.3: Tons of Ratios (15 minutes)

Activity

In the previous activity, students noticed that if right triangles have the same angle measures, they have the same ratios among their side lengths. Students may also have noticed patterns between the columns, such as that when two angles are complementary, there are relationships between the ratios associated with them.

In this activity, students look for patterns in the right triangle table, including patterns that let them make predictions about other data points. Monitor for students who:

  • calculate or estimate the ratio for a 35 degree angle by halving a 70 degree angle
  • calculate or estimate the ratio for a 35 degree angle using the 30 and 40 degree angles
  • use complementary relationships and find the measures of a 35 degree angle using the 55 degree angle
  • draw and measure a right triangle with a 35 degree angle

Asking students what they know about the angle in a triangle with a certain ratio comes up again in the cool-down.

Launch

Display the class table and distribute the blackline master with the completed table. Students will need their completed right triangle table for the next several lessons. Suggest students tape it into their workbook or staple it to their reference chart. They will only need to reference it during this unit.

Ask students, "What do you notice? What do you wonder?" Record and display their responses for all to see. If possible, record the relevant reasoning on or near the table. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

Things students may notice:

  • The angles are increasing by multiples of ten.
  • The values in the opposite \(\div\) hypotenuse column are increasing.
  • The values in the first two ratio columns are all less than one.
  • The first two ratio columns are the same numbers but backwards.

Things students may wonder:

  • Do all complementary angles have the same ratios?
  • What happens if the angle is in between these?
  • What happens if the angle is larger than 90?
Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their reasoning about what they notice and wonder, describe their thinking on a display. Color code connections students make between the ratio columns and complementary angles.
Supports accessibility for: Visual-spatial processing; Conceptual processing

Student Facing

  1. Compare the row for 20 degrees and the row for 70 degrees in the right triangle table. What is the same? What is different?
  2. The row for 55 degrees is given here. Complete the row for 35 degrees. 
    angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
    \(35^\circ\)      
    \(55^\circ\) 0.574 0.819 1.428
  3. What do you know about a triangle with an adjacent leg to hypotenuse ratio value of 0.839?

Student Response

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

Student Facing

Are you ready for more?

  1. What is the range for the possible ratios of each of the following ratios?
    1. adjacent leg \(\div\) hypotenuse
    2. opposite leg \(\div\) hypotenuse
    3. opposite leg \(\div\) adjacent leg
  2. What would the triangle look like if the adjacent leg \(\div\) hypotenuse ratio was 1? Greater than 1?

Student Response

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

Anticipated Misconceptions

If students are struggling to make the connection between the 35 and 55 degree angles prompt them to draw a right triangle with a 35 degree angle.

Activity Synthesis

The goal of this activity is to continue to introduce students to patterns in the right triangle table. Students need not resolve or master concepts like the relationships between complementary angles at this point.

Invite a student who calculated the 35 degree angle ratios by halving the 70 degree angle ratios to share. Tell students, “That's a great conjecture! It seems like if an angle is half the size of another, we could divide the values of the ratios associated with the bigger angle in half to find the values of the ratios associated with the smaller angle. Let’s test that hypothesis using other angles on the chart.” (The conjecture is false, the ratios associated with the 40 degree angle do not have half the value of the ratios associated with the 80 degree angle.)

Invite a student who used the 30 and 40 degree angles to estimate or calculate the 35 degree angles to share their method as a conjecture. “Let’s test that hypothesis using other angles on the chart.” (Yes, at least on this table, if one angle measure is between two other angle measures, then the values of the associated ratios will be between the values of the ratios associated with the larger and smaller angles. No, the values of the ratios associated with a 50 degree angle are not exactly the midpoint of the values of the ratios associated with a 40 and a 60 degree angle.)

Invite a student who used the complementary relationship between 55 degrees and 35 degrees to reason about the ratios of the adjacent side to the hypotenuse or the opposite side to the hypotenuse to share their method as a conjecture. “Let’s test that hypothesis using other angles on the chart.” (\(20 + 70 = 90\), and for the 20 degree and 70 degree angles, the values of the ratios in the first two columns are swapped, so it seems like the same pattern would work here.) Remind students that we define complementary angles as two angles whose measures add up to 90 degrees.

Invite a student who made a right triangle with a 35 degree angle to confirm which solution methods matched their measurements.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares how they calculated the 35 degree angle ratios, invite the class to respond using one of the following sentence frames: “I agree because . . .”, “I disagree because . . .” or "I have a question about . . . ." Press for mathematical language and details by asking students to elaborate on an idea or to provide a supporting example or counterexample. 
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Display the class right triangle table. In the cool-down and in the subsequent lessons, students will apply the data in the table to estimate unknown angle measures, and to find unknown side lengths in right triangles.

Let students know that people have made tables like they are making for thousands of years in Africa, Greece, the Middle East, India, and other parts of the world. Tables like this are used today in computer graphics programs.

Students will use the right triangle table to think about unknown angles. They will need to use patterns that they noticed and studied today. Tell students you have information about another row in the right triangle table, but the angle is missing. Show students this data and ask them to estimate the angle measures.

angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
  0.421 0.903 2.145

Students should use what they know about the increasing and decreasing patterns in the right triangle table to place this row between 60 and 70 degrees.

angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
  0.903 0.421 0.466

Students should use what they know about the increasing and decreasing patterns in the right triangle table to place this row between 20 and 30 degrees and to notice that the angle represented by this row is complementary to the angle represented by the previous row.

For students struggling with determining an angle from a ratio, emphasize estimation. “What’s an angle that you know is too high? What’s an angle that you know is too low? Explain how you know.”

4.4: Cool-down - Lift Off (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 angles are similar to each other. This means that the ratios of corresponding side lengths are equal for all right triangles with the same acute angles.

Three right triangles, On left triangle ABC, in middle triangle DEF, on right triangle LMN.

These triangles are all similar by the Angle-Angle Triangle Similarity Theorem. Focusing on the 25 degree angles, we see that all 3 triangles have adjacent leg to hypotenuse ratios of approximately 0.91.

Because all right triangles with the same acute angle measures have the same ratios, we can look for patterns that will help us solve problems. The right triangle table comes from measuring and finding ratios in several right triangles with different angle measures.

angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
\(25^\circ\) 0.906 0.423 0.466
\(35^\circ\) 0.819 0.574 0.700
\(45^\circ\) 0.707 0.707 1.000
\(55^\circ\) 0.574 0.819 1.428
\(65^\circ\) 0.423 0.906 2.145

Some ratios in this table are repeated. Notice that the rows for 25 degrees and 65 degrees have 2 of the same ratios. What is special about 25 and 65? They are complementary angles, that is, the 2 angles sum to 90 degrees. This seems to be true for other complementary angles. Notice that \(35+55=90\) and those rows both have 0.819 as a ratio.