Lesson 3

Construction Techniques 1: Perpendicular Bisectors

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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.

two intersecting circles
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:

\(CB+BD > CD\)

Solution

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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.

\(AB \perp CD\)

Two intersecting line segments.

 

Solution

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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.

Two congruent circles centered at A and B, each pass through the center of the other, intersecting at C. Radii AB, AC and BC form an equilateral triangle.

Solution

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(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.

Two congruent circles, centered at C and D, each passes through the center of the other. There is a vertical line segment between their intersection points, labeled A and B.
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 \(B\).

Solution

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(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.

Two circles intersecting.
A:

\(AB=BC\)

B:

\(AB=BD\)

C:

\(AD=2AC\)

D:

\(BC=CD\)

E:

\(BD=CD\)

Solution

For access, consult one of our IM Certified Partners.

(From Unit 1, Lesson 1.)