Lesson 3

Half an Equilateral Triangle

3.1: Notice and Wonder: Triangle Slices (5 minutes)

Warm-up

In this activity, students first sketch the situation and then notice and wonder about their sketches. The purpose of this warm-up is to elicit the idea that equilateral triangles and altitudes have properties we can generalize, which will be useful when students look for patterns in a later activity. While students may notice and wonder many things about their sketch, angle measures and the proportions of side lengths are the important discussion points.

Launch

If students have printed workbooks invite them to sketch on scrap paper rather than in their workbooks, as the surrounding information may influence their observations.

Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

Sketch an equilateral triangle and an altitude from any vertex in the equilateral triangle.

What do you notice? What do you wonder?

Student Response

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

Anticipated Misconceptions

If needed, remind students of the definition of altitude: “An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.”

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near a displayed sketch. 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.

3.2: Decomposing Equilateral Triangles (15 minutes)

Optional activity

In a previous lesson, students proved that all equilateral triangles are similar. In this lesson, they convince themselves that equilateral triangles decomposed into two congruent halves are similar right triangles, and therefore the ratios of their side lengths will be equal.

The first triangle students encounter in this activity is an equilateral triangle with side lengths of two, which means that the shortest side of the 30-60-90 triangles formed is one unit long. Monitor for students who use scale factors to generate side lengths in the triangles they measure as well as students who measure and compute all the ratios.

Students will need a set of equilateral triangles to measure. They could use isometric dot paper, construction tools, or the blackline master.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Distribute equilateral triangles of several different sizes or the tools to make them to each group.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

  1. Here is an equilateral triangle with side length 2 units and an altitude drawn. Find the values of \(x\) and \(y\).

    Triangle. 
  2. Measure several more of these “half equilateral triangles” by drawing equilateral triangles and altitudes. Compute the ratios of the side lengths of these new triangles.
  3. Make a conjecture about side lengths in “half equilateral triangles.”

Student Response

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

Anticipated Misconceptions

If students are struggling to organize their thinking, suggest that they make a table. Help students brainstorm what would be good columns to organize their measurements and calculations, such as “short leg length,” “altitude length,” and “ratio of hypotenuse to short leg.”

Activity Synthesis

Ask students what they found for the ratio of the altitude, \(y\), to the short leg, \(x\). Include students who approximated the lengths using a calculator and students who left the length of the altitude as \(\sqrt3\).

Ask students why the altitude in any equilateral triangle seems to be half the side length multiplied by about 1.7 each time. (The altitude is also the median in an equilateral triangle, so the short leg is half the side length. All equilateral triangles are similar to the first one we studied, where the altitude was \(\sqrt3\).)

Display the image and ask students if they agree that both of these things are true:

  • The length \(y\) is about 1.7 times \(x\).
  • The length \(y\) is exactly \(\sqrt3\) times \(x\).
A 30-60-90 triangle.
Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing their equilateral triangles and the ratios of the side lengths. Allow students time to quietly circulate and analyze the conjectures and justifications in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight the different ways students found the ratio of the altitude to the short leg. This will help students make connections between the ratio of the altitude to the short leg, \(\sqrt{3}\), and the decimal approximation 1.7.
Design Principle(s): Optimize output; Cultivate conversation

3.3: Generalize Half Equilateral Triangles (15 minutes)

Optional activity

In the previous activity and summary, students generalized that the altitudes of equilateral triangles are related to half the side length of the triangle (the short leg) by a factor of about 1.7, or exactly \(\sqrt3\).

In this activity, students apply their generalization about the altitudes of equilateral triangles to 30-60-90 triangles. To find the lengths of the unlabeled sides in the second figure, students will need to generalize that a right triangle with one 60 degree angle is a 30-60-90 triangle (because it’s half an equilateral triangle, or because of the Triangle Angle Sum Theorem, or because it’s similar to the triangle in the first figure).

Monitor for different solutions in the third figure. Students will have to use the ratios to find unknown side lengths. Students may use the approximate ratio \(1.7:1:2\) or the exact ratio \(\sqrt3:1:2\).

Launch

Action and Expression: Internalize Executive Functions. Provide students with a three-column table to organize. Use these column headings: the side adjacent to the 60 degree angle, the side opposite to the 60 degree angle, and the hypotenuse. The table will provide a visual support for students to identify the type of triangle and pattern.
Supports accessibility for: Language; Organization

Student Facing

Three triangles. 

Calculate the lengths of the 5 unlabeled sides.

Student Response

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

Student Facing

Are you ready for more?

Here is a collection of triangles which all have angles measuring 30, 60 and 90 degrees.

collection of 5 triangles which all have angles measuring 30, 60 and 90 degrees. Base length of largest triangle = 3 point 3 units. Each subsequent triangle’s base = shortest side of previous triangle.

What is the total area enclosed by the 5 triangles?

Student Response

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

Activity Synthesis

Make sure all students understand that the three triangles are 30-60-90 right triangles, and represent half of an equilateral triangle, and so are prepared to connect their reasoning from earlier activities to this activity.

Focus on the third figure. Ask students what was different about that figure that might have made the task a bit harder. (There's only one side given and it has a square root in it.) Invite previously selected students to explain their strategies for calculating the lengths of the other two sides.

Lesson Synthesis

Lesson Synthesis

Create a class display of half an equilateral triangle to accompany the display of half a square. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like:

A 45-45-90 triangle.
A 30-60-90 triangle.

Ask students what they notice and what they wonder about the images. (There’s two triangles. One is isosceles. One is half a square. One is half an equilateral triangle. They both have square roots. Why are we looking at two triangles? Are the triangles related? What are we going to do with these?)

If applicable, remind students that this type of image might be provided as a reference on standardized tests. Ask students how they might decide what types of problems to use it on, and how they would solve those problems. Here are three sample problems to discuss:

  1. The side length of an equilateral triangle is 8 units. What is the height of the triangle?
  2. The diagonal of a square is \(5\sqrt2\) units. What is the side length of the square?
  3. An engineer is standing 10 meters away from a building. When she looks up at the top of the building, she measures the angle from the ground to the top of the building to be \(60^{\circ}\). What is the height of the building?

Students may recognize the height of an equilateral triangle divides it in half to match the image. Similarly, the diagonal of a square divides it in half to match that image. Students may need to sketch the final sample problem to recognize that it matches the image. Note that they could use the Pythagorean Theorem to solve all three problems, since that is where the labels on the image come from.

3.4: Cool-down - Half of the Half (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Drawing the altitude of an equilateral triangle decomposes the equilateral triangle into 2 congruent triangles. They are right triangles with acute angles of 30 and 60 degrees. These congruent angles make all triangles with angles 30, 60, and 90 degrees similar by the Angle-Angle Triangle Similarity Theorem.

If we consider a right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side 1 unit long, then the hypotenuse must be 2 units long since the triangle can be thought of as half of an equilateral triangle. Call the length of the altitude \(a\). By the Pythagorean Theorem, we can say \(a^2+1^2=2^2\) so \(a=\sqrt3\).

Vertical segment drawn down equilateral triangle forming 2 right triangles. Right triangle legs are 1 and square root 3. Hypotenuse is 2. Other angles are 60 and 30 degrees. 60 adjacent to side of 1.

Now, consider another right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side \(y\) units long. By the Angle-Angle Triangle Similarity Theorem, it must be similar to the right triangle with angles 30, 60, and 90 degrees and with sides 1, \(\sqrt3\), and 2 units long. The scale factor is \(y\), so a triangle with angles 30, 60, and 90 degrees has side lengths \(y, y\sqrt3,\) and \(2y\) units long.

Triangle. 
Triangle A B C.
Triangle E G H.

In triangle \(ABC, 2y=5\) so \(y=\frac 52\). That means \(DB\) is \(\frac 52\) units and \(DC\) is \(\frac 52 \sqrt 3\) units.

In triangle \(EGH, y\sqrt 3=4 \) so \(y=\frac{4}{\sqrt 3}\). That means \(FG\) is \(\frac{4}{\sqrt 3}\) units and \(EG\) is \(2 \frac{4}{\sqrt 3}\) or \(\frac{8}{\sqrt 3}\) units.