Lesson 4

Construction Techniques 2: Equilateral Triangles

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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. Explain how we know triangle \(ABC\) is equilateral.

Two congrent circles intersect. Left circle center A, right circle center b. Cirlces intersect at point C. Segments A B, B C and A C are drawn.

Solution

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Problem 2

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

Three intersecting circles.

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. Select all the true statements.

Two congruent intersecting circles, each pass through the other’s center at points A and B. Circles intersect at top point C and bottom point D. Line segments A B, C D, A C, B C, A D,  BD are drawn.
A:

\(AC=BC\)

B:

\(AC=BD\)

C:

\(CD=AB\)

D:

\(ABCD\) is a square.

E:

\(ABD\) is an equilateral triangle.

F:

\(CD=AB+AB\)

Solution

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Problem 4

Line segment \(CD\) is the perpendicular bisector of line segment \(AB\). Is line segment \(AB\) the perpendicular bisector of line segment \(CD\)?

Line segment CD as perpendicular bisector of line segment AB intersecting at point M.

Solution

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(From Unit 1, Lesson 3.)

Problem 5

Here are 2 points in the plane.

Two points, A and B.
  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.

Solution

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(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.

\(AB \perp CD\)

Two intersecting line segments.
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:

\(A M > B M\)

Solution

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(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 \(AC\).

Three congruent circles, each pass through the others center at points A, B, and E, and intersect line segment with the endpoints C and D. Points C, A, B, E, and D are collinear and equidistant.

Solution

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(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.

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 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 \(AB\)?

Diagram of circles and quadrilateral C D B E.
A:

\(CB\)

B:

\(CD\)

C:

\(CE\)

D:

\(CA\)

Solution

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(From Unit 1, Lesson 2.)