# Construction Techniques 5: Squares

ID: datam-nanod Illustrative Math
Subject: Geometry

# Construction Techniques 5: Squares

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##### Problem 1

1) Which of these statements is true?

a) $\text{All rectangles are regular polygons. }$b) $\text{All squares are regular polygons. }$c) $\text{All rhombi are regular polygons. }$d) $\text{All parallelograms are regular polygons.}$
##### Problem 2

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. ##### Problem 3

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) $\text{Construct an equilateral triangle with side } CD \text{. }$b) $\text{Construct a line through point } B \text{ perpendicular to } AB \text{. }$c) $\text{Construct a segment with the same length as } AB \text{ with endpoint } C \text{. }$d) $\text{Construct a line through point } C \text{ perpendicular to } CD \text{.}$
##### Problem 4

4) Jada wanted to construct a line perpendicular to line ﻿$l$﻿ through point ﻿$C$﻿. The diagram shows her construction. What was her mistake? ##### Problem 5

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? ##### Problem 6

6) Here is a straightedge and compass construction. Use a straightedge to draw an equilateral triangle on the figure. 7) Explain how you know the triangle is equilateral.

##### Problem 7

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