Lesson 1 Under Construction Develop Understanding

Jump Start

Examine the diagram. (Note: points $M$ and $N$ are at the center of their respective circles.)

1.

Write as many equality statements as you can to represent equal lengths in the diagram.

2.

Write as many congruence statements as you can to represent congruent segments in the diagram.

3.

Be prepared to state how you know which segments are congruent. Are there any segments you are wondering about, but aren’t sure if they are really congruent?

Learning Focus

Construct a rhombus, a perpendicular bisector, and a square using only a compass and a straightedge (unmarked ruler) as tools.

How do I use geometric objects, such as circles and lines, to construct geometric figures like rhombuses and squares, rather than using measurement tools, such as rulers and protractors, to draw such figures?

Open Up the Math: Launch, Explore, Discuss

In ancient times, one of the only tools builders and surveyors had for laying out a plot of land or the foundation of a building was a piece of rope.

There are two geometric figures you can create with a piece of rope: you can pull it tight to create a line segment, or you can fix one end, and—while extending the rope to its full length—trace out a circle with the other end. Geometric constructions have traditionally mimicked these two processes using an unmarked straightedge to create a line segment and a compass to trace out a circle (or sometimes a portion of a circle called an arc). Using only these two tools, you can construct all kinds of geometric shapes.

Suppose you want to construct a rhombus using only a compass and straightedge. You might begin by drawing a line segment to define the length of a side, and drawing another ray from one of the endpoints of the line segment to define an angle, as in the sketch.

Now the hard work begins. We can’t just keep drawing line segments because we have to be sure that all four sides of the rhombus are the same length. We have to stop drawing and start constructing.

Construct a Rhombus

Knowing what you know about circles and line segments, how might you locate point $C$ on the ray in the diagram given, so the distance from $B$ to $C$ is the same as the distance from $B$ to $A$ ?

1.

Describe how you will locate point $C$ and how you know $\overset{―}{B C} ≅ \overset{―}{B A}$ , then construct point $C$ on the diagram given.

Now that we have three of the four vertices of the rhombus, we need to locate point $D$ , the fourth vertex.

2.

Describe how you will locate point $D$ and how you will know $\overset{―}{C D} ≅ \overset{―}{D A} ≅ \overset{―}{A B}$ , then construct point $D$ on the diagram.

Pause and Reflect

Construct a Perpendicular Bisector of a Segment and a Square (a rhombus with right angles)

The only difference between constructing a rhombus and constructing a square is that a square contains right angles. Therefore, we need a way to construct perpendicular lines using only a compass and a straightedge.

We will begin by inventing a way to construct a perpendicular bisector of a line segment.

3.

Given $\overset{―}{R S}$ , fold and crease the paper so that point $R$ is reflected onto point $S$ . Based on the definition of reflection, what do you know about this “crease line”?

You have “constructed” a perpendicular bisector of $\overset{―}{R S}$ by using a paper-folding strategy. Is there a way to construct this line using a compass and a straightedge?

4.

Experiment with the compass to see if you can develop a strategy to locate points on the “crease line.” When you have located at least two points on the “crease line,” use the straightedge to finish your construction of the perpendicular bisector. Describe your strategy for locating points on the perpendicular bisector of $\overset{―}{R S}$ .

Now that you have created a line perpendicular to $\overset{―}{R S}$ , we will use the right angle formed to construct a square.

5.

Label the midpoint of $\overset{―}{R S}$ on the diagram as point $M$ . Using $\overset{―}{M S}$ as one side of the square, and the right angle formed by $\overset{―}{S M}$ and the perpendicular line drawn through point $M$ as the beginning of a square, finish constructing this square on the diagram. (Hint: Remember that a square is also a rhombus, and you have already constructed a rhombus in the first part of this task.)

Ready for More?

Draw a line and select two arbitrary points on the line. Treating the line segment between the two points you selected as one side of a square, use the construction strategies you invented in today’s task to construct the square. Demonstrate your thinking by showing all of the circles, or portions of circles, you use in your construction.

Takeaways

I used circles and lines as a construction tool today.

Circles are useful construction tools because
Congruent circles are useful construction tools because
The congruent circles in the construction of the perpendicular bisector helped me to notice that

Vocabulary

congruence statement
construction
equality statements
radii
ray
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about constructions: creating geometric figures precisely, using only a compass and a straightedge. Using only these tools, we constructed a rhombus with a given side and angle, constructed the perpendicular bisector of a side, and constructed a square with a given right angle and line segment for a side. We learned the value of the definition of a circle—the set of all points in a plane equidistant from a fixed center point—since circles allow us to construct congruent line segments.

Retrieval

1.

Figure $A B C D$ is a rhombus.

Use a straightedge to draw the two diagonals.
Using a compass, construct a circle with center at point $A$ and a radius of length $A B$ . Then construct a circle at point $C$ with radius length $C D$ .
What do you notice about the intersections of the two circles?

2.

Solve the system of equations. Choose an appropriate method: graphing, substitution, or elimination.

${\begin{cases} y = 6 x + 4 \\ 9 x + y = - 1 \end{cases}$