# Lesson 3

Construction Techniques 1: Perpendicular Bisectors

• Let’s explore equal distances.

### 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. Select all the true statements.

A:

Line $$CD$$ is perpendicular to segment $$AB$$

B:

Point $$M$$ is the midpoint of segment $$AB$$

C:

The length $$AB$$ is the equal to the length $$CD$$.

D:

Segment $$AM$$ is perpendicular to segment $$BM$$

E:

$$CB+BD > CD$$

### Problem 2

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. Is point $$E$$ closer to point $$A$$, closer to point $$B$$, or the same distance between the points? Explain how you know.

### Problem 3

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

(From Unit 1, Lesson 2.)

### Problem 4

This diagram was created by starting with points $$C$$ and $$D$$ and using only straightedge and compass to construct the rest. All steps of the construction are visible. Select all the steps needed to produce this diagram.

A:

Construct a circle centered at $$A$$.

B:

Construct a circle centered at $$C$$.

C:

Construct a circle centered at $$D$$.

D:

Label the intersection points of the circles $$A$$ and $$B$$.

E:

Draw the line through points $$C$$ and $$D$$.

F:

Draw the line through points $$A$$ and $$B$$.

(From Unit 1, Lesson 2.)

### Problem 5

This diagram was constructed with straightedge and compass tools. $$A$$ is the center of one circle, and $$C$$ is the center of the other. Select all true statements.

A:

$$AB=BC$$

B:

$$AB=BD$$

C:

$$AD=2AC$$

D:

$$BC=CD$$

E:

$$BD=CD$$

(From Unit 1, Lesson 1.)