Lesson 3

Half an Equilateral Triangle

  • Let’s investigate the properties of altitudes of equilateral triangles.

Problem 1

Select all statements that are true about equilateral triangle \(ABC\).

Triangle A B C.
A:

Angles \(B\) and \(C\) are 60 degrees.

B:

\(x = 3\sqrt{3}\)

C:

\(x = 6\sqrt{3}\)

D:

Triangle \(ABD\) is congruent to triangle \(ACD\).

E:

\(BD\) and \(CD\) are both 3 units long.

Problem 2

Find the length of each leg.

Triangle A B C. Angle C is labeled 30 degrees. Angle A is labeled 60 degrees. Angle B is marked with a right angle symbol. Side A C is labeled 10.

Problem 3

An equilateral triangle has a side length of 10 units. What is its area?

Problem 4

Find the lengths of the legs.

Right triangle. Hypotenuse is 10. One of the non-right angles is labeled 45 degrees.
(From Unit 4, Lesson 2.)

Problem 5

A square has side length 3 units. What is the length of the diagonal? 

A:

3 units

B:

\(\frac{3}{\sqrt2}\) units

C:

\(3 \sqrt2\) units

D:

6 units

(From Unit 4, Lesson 2.)

Problem 6

A step has a height of 5 inches. A ramp starts 4 feet away from the base of the step, making a \(5.9^\circ\) angle with the ground. What can you say about the angle the ramp would make with the ground if the ramp starts farther away from the step? 

A:

The angle would decrease.

B:

The angle would remain the same.

C:

The angle would increase.

D:

We cannot determine anything about the angle.

(From Unit 4, Lesson 1.)

Problem 7

Segment \(A’B’\) is parallel to segment \(AB\).

  1. What is the length of segment \(A'B'\)?
  2. What is the length of segment \(B’B\)?
Triangle A prime B prime C is set inside triangle A B C.
(From Unit 3, Lesson 11.)

Problem 8

Here is triangle \(POG\). Match the description of the rotation with the image of \(POG\) under that rotation.

Triangle P O G on an isometric grid. Side G O is located on a vertical grid line, and P is located horizontally to the right of G.

Image 1

Triangle P prime O prime G prime on isometric grid.

Image 2

Triangle P prime O prime G prime. Side P prime O prime is vertical on the grid, with P prime below O prime. G prime is located down and to the right from O prime.

Image 3

Triangle P prime O prime G prime. Side P prime O prime is vertical on the grid, with O prime below P prime. G prime is located up and to the left from O prime.

Image 4

Triangle P prime O prime G prime. Side P prime O prime is on a slanted grid line with P prime located down and to the right from O prime. G prime is located up and to the right from O prime.
A:

Rotate 60 degrees counterclockwise around \(P\).

B:

Rotate 60 degrees clockwise around \(O\).

C:

Rotate 120 degrees clockwise around \(G\).

D:

Rotate 60 degrees clockwise around \(G\).

1:

Image 1

2:

Image 2

3:

Image 3

4:

Image 4

(From Unit 1, Lesson 13.)