Unit 4, Lesson 3
Half an Equilateral Triangle
Geometry
3.2
Activity
Decomposing Equilateral Triangles
-
Here is an equilateral triangle with side length 2 units and an altitude drawn. Find the values of
and . 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 bottom left angle is labeled 60 degrees. The right side is labeled 2. The left side is labeled 2. The vertical line is labeled y. Both angles, split by the vertical line, at the top are labeled 30 degrees. The horizontal side, right of the vertical line to the angle is labeled x. - Compute the quotients:
- length of the hypotenuse (2) divided by length of the short leg (
) - length of the altitude (
) divided by length of the short leg ( )
- length of the hypotenuse (2) divided by length of the short leg (
-
Measure several more of these “half equilateral triangles” by drawing equilateral triangles and altitudes. Compute the same quotients for each right triangle created:
- length of the hypotenuse divided by length of the short leg
- length of the altitude divided by length of the short leg
- Make a conjecture about side lengths in “half equilateral triangles.”
3.3
Activity
Generalize Half Equilateral Triangles
Calculate the lengths of the 5 unlabeled sides.
Triangle A B C. Angle A is a right angle. Angle B is labeled 60 degrees. Side A B is labeled 5. Side B C is labeled 10. Triangle F G H. Angle F is a right angle. Angle G is labeled 60 degrees. Side F G is labeled 3. Triangle L M N. Angle L is a right angle. Angle M is labeled 60 degrees. Side N L is labeled 2 times square root of 3.
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
Now, consider another right triangle with angle measures of 30, 60, and 90 degrees, and with the shortest side
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
In triangle
None
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