Lesson 4

Construction Techniques 2: Equilateral Triangles

• Let’s identify what shapes are possible within the construction of a regular hexagon.

Problem 1

This diagram is a straightedge and compass construction. $$A$$ is the center of one circle, and $$B$$ is the center of the other. Explain how we know triangle $$ABC$$ is equilateral.

Problem 2

$$A$$, $$B$$, and $$C$$ are the centers of the 3 circles. How many equilateral triangles are there in this diagram?

Problem 3

This diagram is a straightedge and compass construction. $$A$$ is the center of one circle, and $$B$$ is the center of the other. Select all the true statements.

A:

$$AC=BC$$

B:

$$AC=BD$$

C:

$$CD=AB$$

D:

$$ABCD$$ is a square.

E:

$$ABD$$ is an equilateral triangle.

F:

$$CD=AB+AB$$

Problem 4

Line segment $$CD$$ is the perpendicular bisector of line segment $$AB$$. Is line segment $$AB$$ the perpendicular bisector of line segment $$CD$$?

(From Unit 1, Lesson 3.)

Problem 5

Here are 2 points in the plane.

1. Using only a straightedge, can you find points in the plane that are the same distance from points $$A$$ and $$B$$? Explain your reasoning.
2. Using only a compass, can you find points in the plane that are the same distance from points $$A$$ and $$B$$? Explain your reasoning.
(From Unit 1, Lesson 3.)

Problem 6

In this diagram, line segment $$CD$$ is the perpendicular bisector of line segment $$AB$$. Assume the conjecture that the set of points equidistant from $$A$$ and $$B$$ is the perpendicular bisector of $$AB$$ is true. Select all statements that must be true.

A:

$$A M = B M$$

B:

$$C M = D M$$

C:

$$E A = E M$$

D:

$$E A < E B$$

E:

$$A M < A B$$

F:

$$A M > B M$$

(From Unit 1, Lesson 3.)

Problem 7

The diagram was constructed with straightedge and compass tools. Name all segments that have the same length as segment $$AC$$.

(From Unit 1, Lesson 1.)

Problem 8

Starting with 2 marked points, $$A$$ and $$B$$, precisely describe the straightedge and compass moves required to construct the quadrilateral $$ACBD$$ in this diagram.

(From Unit 1, Lesson 2.)

Problem 9

In the construction, $$A$$ is the center of one circle and $$B$$ is the center of the other. Which segment has the same length as $$AB$$?

A:

$$CB$$

B:

$$CD$$

C:

$$CE$$

D:

$$CA$$

(From Unit 1, Lesson 2.)