Lesson 7

Construction Techniques 5: Squares

Problem 1

Which of these statements is true?

A:

All rectangles are regular polygons.

B:

All squares are regular polygons.

C:

All rhombi are regular polygons.

D:

All parallelograms are regular polygons.

Solution

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Problem 2

This diagram is a straightedge and compass construction of a square $$BACD$$ (not all markings are shown). The construction followed these steps:

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. Use a compass to construct a circle centered at $$A$$ passing through $$B$$
6. Label an intersection point of that circle and the line from step 3 as $$C$$
7. Use a previous construction to construct a line parallel to $$AB$$ passing through $$C$$
8. Label the intersection of that line and the line from step 4 as $$D$$
9. Use a straightedge to construct the segments $$AC$$, $$CD$$, and $$DB$$

Explain why you need to construct a circle in step 5.

Solution

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Problem 3

To construct a line passing through the point $$C$$ that is parallel to the line $$AB$$, the first step is to create a line through $$C$$ perpendicular to $$AB$$. What is the next step?

A:

Construct an equilateral triangle with side $$CD$$.

B:

Construct a line through point $$B$$ perpendicular to $$AB$$.

C:

Construct a segment with the same length as $$AB$$ with endpoint $$C$$.

D:

Construct a line through point $$C$$ perpendicular to $$CD$$.

Solution

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(From Unit 1, Lesson 6.)

Problem 4

Jada wanted to construct a line perpendicular to line $$\ell$$ through point $$C$$. The diagram shows her construction. What was her mistake?

Solution

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(From Unit 1, Lesson 6.)

Problem 5

Noah is trying to bisect angle $$BAC$$. He draws circles of the same radius with centers $$B$$ and $$C$$ and then uses one of the points of intersection for his ray. What mistake has Noah made in his construction?

Solution

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(From Unit 1, Lesson 5.)

Problem 6

Here is a straightedge and compass construction. Use a straightedge to draw an equilateral triangle on the figure. Explain how you know the triangle is equilateral.

Solution

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(From Unit 1, Lesson 4.)

Problem 7

Here are 2 points in the plane.  Explain how to construct a line segment that is half the length of segment $$AB$$.