Unit 3, Lesson 3
Half an Equilateral Triangle
Integrated Math 2
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 this:
Ask students what they notice and what they wonder about the images. (There are two triangles. One is isosceles and half a square. The other 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 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 . What is the height of the building?
Students may recognize that the image of the 30-60-90 triangle results when the height of an equilateral triangle divides the triangle in half. Similarly, the image of the 45-45-90 triangle results when the diagonal of a square divides the square in half. 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.
Student Lesson Summary
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 . By the Pythagorean Theorem, we can say , so .
Now, consider another right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side 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, , and 2 units long. The scale factor is , so a triangle with angles 30, 60, and 90 degrees has side lengths and units long.
Triangle with a vertex above bottom horizontal side. A vertical line drawn from top vertex to horizontal side forming a right angle. The bottom right angle is labeled 60 degrees. The right side is labeled 2 y. The vertical line is labeled y times square root of 3. The right side top angle is labeled 30 degrees. The horizontal side, right of the vertical line to the angle is labeled y.
Triangle A B C. Point D lies on A B. A line is drawn from point C to point D with a right angle symbol as it hits point D. Side B C is labeled 5. Angle C B A is labeled 60 degrees. Angle B C D is labeled 30 degrees.
Triangle E G H. Point F lies on G H. A line is drawn from point E to point F with a right angle symbol as it hits point F. E F is labeled 4. Angle E G H is labeled 60 degrees. Angle G E F is labeled 30 degrees.
In triangle , so . That means is units and is units.
In triangle , so . That means is units and is , or , units.