Lesson 3
Half an Equilateral Triangle
3.1: Notice and Wonder: Triangle Slices (5 minutes)
Warmup
In this activity, students first sketch the situation and then notice and wonder about their sketches. The purpose of this warmup 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 wholeclass 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 306090 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.
Supports accessibility for: Language; Socialemotional skills
Student Facing

Here is an equilateral triangle with side length 2 units and an altitude drawn. Find the values of \(x\) and \(y\).
 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.
 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 \(x\).
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 306090 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 306090 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
Supports accessibility for: Language; Organization
Student Facing
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.
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 306090 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:
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:
 The side length of an equilateral triangle is 8 units. What is the height of the triangle?
 The diagonal of a square is \(5\sqrt2\) units. What is the side length of the square?
 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: Cooldown  Half of the Half (5 minutes)
CoolDown
Cooldowns 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 AngleAngle 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 \(a=\sqrt3\).
Now, consider another right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side \(y\) units long. By the AngleAngle 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.
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.