Lesson 5

Construction Techniques 3: Perpendicular Lines and Angle Bisectors

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Problem 1

This diagram is a straightedge and compass construction of a line perpendicular to line \(AB\) passing through point \(C\). Explain why it was helpful to construct points \(D\) and \(A\) to be the same distance from \(C\).

Three circles.

Solution

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Problem 2

This diagram is a straightedge and compass construction.

Select all true statements.

Three circles
A:

Line \(EF\) is the bisector of angle \(BAC\).

B:

Line \(EF\) is the perpendicular bisector of segment \(BA\).

C:

Line \(EF\) is the perpendicular bisector of segment \(AC\).

D:

Line \(EF\) is the perpendicular bisector of segment \(BD\).

E:

Line \(EF\) is parallel to line \(CD\).

Solution

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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. A rhombus is a quadrilateral with 4 congruent sides. Explain why quadrilateral \(ACBD\) is a rhombus.

Two congruent circles, with centers A and B, each pass through the center of the other and intersect at C and D. Radii AC, CB, DA and BD form quadrilateral ACBD.

Solution

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(From Unit 1, Lesson 4.)

Problem 4

\(A\), \(B\), and \(C\) are the centers of the three circles. Which line segment is congruent to \(HF\)?

Three intersecting circles.
A:

\(AB\)

B:

\(CD\)

C:

\(DF\)

D:

\(CB\)

Solution

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(From Unit 1, Lesson 4.)

Problem 5

In the construction, \(A\) is the center of one circle, and \(B\) is the center of the other. Explain why segment \(EA\) is the same length as segment \(BC\).

A diagram of circles and line segments.

Solution

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(From Unit 1, Lesson 2.)

Problem 6

\(AB \perp CD\)

Two intersecting line segments.

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 \(M\) closer to point \(A\), closer to point \(B\), or the same distance from both points? Explain how you know.

Solution

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(From Unit 1, Lesson 3.)

Problem 7

A sheet of paper with points \(A\) and \(B\) is folded so that \(A\) and \(B\) match up with each other. 

Two points, A and B.

Explain why the crease in the sheet of paper is the perpendicular bisector of segment \(AB\). (Assume the conjecture that the set of points equidistant from \(A\) and \(B\) is the perpendicular bisector of segment \(AB\) is true.)

Solution

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(From Unit 1, Lesson 3.)

Problem 8

Here is a diagram of a straightedge and compass construction. \(C\) is the center of one circle, and \(B\) is the center of the other. Explain why the length of segment \(CB\) is the same as the length of segment \(CD\).

Two circles, one large centered at point C. Points A, B, and D lie on the large circle. The smaller circle centered at point B intersects larger circle at points A and D. Radii CA, CB, and CD are drawn. Radii BA and BD are drawn.

Solution

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(From Unit 1, Lesson 1.)