Lesson 9

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Problem 1

Which construction can be used to determine whether point \(C\) is closer to point \(A\) or point \(B\)?

A:

Construct triangle \(ABC\).

B:

Construct a line perpendicular to segment \(AB\) through point \(C\).

C:

Construct the bisector of angle \(ACB\).

D:

Construct the perpendicular bisector of segment \(AB\).

Solution

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Problem 2

The diagram is a straightedge and compass construction. Lines \(\ell\), \(m\), and \(n\) are the perpendicular bisectors of the sides of triangle \(ABC\). Select all the true statements.

Triangle A B C with intersecting lines. 
A:

Point \(E\) is closer to point \(A\) than it is to point \(C\).

B:

Point \(L\) is closer to point \(B\) than it is to point \(A\).

C:

Point \(D\) is closer to point \(B\) than it is to point \(C\).

D:

Point \(J\) is closer to point \(A\) than it is to point \(B\) or point \(C\).

E:

Point \(K\) is closer to point \(C\) than it is to point \(A\) or point \(B\).

F:

Point \(L\) is closer to point \(C\) than it is to point \(A\) or point \(B\).

Solution

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Problem 3

Decompose the figure into regions that are closest to each vertex. Explain or show your reasoning.

Two triangles, creating Quadrilateral D C A B. Triangle B C D on the left, Triangle A B C on the right. Both triangles share side B C.

Solution

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Problem 4

Which construction could be used to construct an isosceles triangle \(ABC\) given line segment \(AB\)?

A:

Mark a third point \(C\) not on segment \(AB\). Draw segments \(AC\) and \(BC\).

B:

Label a point \(C\) on segment \(AB\) and construct a line perpendicular to \(AB\) through point \(C\). Draw segments \(AC\) and \(BC\).

C:

Construct the perpendicular bisector of segment \(AB\). Mark the intersection of this line and \(AB\) and label it \(C\). Draw segments \(AC\) and \(BC\).

D:

Construct the perpendicular bisector of segment \(AB\). Mark any point \(C\) on the perpendicular bisector except where it intersects \(AB\). Draw segments \(AC\) and \(BC\).

Solution

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Problem 5

Select all true statements about regular polygons.

A:

All angles are right angles.

B:

All angles are congruent.

C:

All side lengths are equal.

D:

There are exactly 4 sides.

E:

There are at least 3 sides.

Solution

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(From Unit 1, Lesson 7.)

Problem 6

This diagram shows the beginning of a straightedge and compass construction of a rectangle.

The construction followed these steps:

Three lines. Two vertical lines intersect a horizontal line at points A and B, making right angles. The horizontal line is on the bottom of the diagram. On the left line, point C is marked at the top.
  1. Start with two marked points \(A\) and \(B\)
  2. Use a straightedge to construct line \(AB\)
  3. Use a previous construction to construct a line perpendicular to \(AB\) passing through \(A\)
  4. Use a previous construction to construct a line perpendicular to \(AB\) passing through \(B\)
  5. Mark a point \(C\) on the line perpendicular to \(AB\) passing through \(A\)

Explain the steps needed to complete this construction.

Solution

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(From Unit 1, Lesson 7.)

Problem 7

This diagram is a straightedge and compass construction. Is it important that the circle with center \(B\) passes through \(D\) and that the circle with center \(D\) passes through \(B\)? Show or explain your reasoning.

Three circles

Solution

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(From Unit 1, Lesson 5.)