Lesson 1 Under Construction Develop Understanding
Examine the diagram. (Note: points
Write as many equality statements as you can to represent equal lengths in the diagram.
Write as many congruence statements as you can to represent congruent segments in the diagram.
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?
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
Describe how you will locate point
Now that we have three of the four vertices of the rhombus, we need to locate point
Describe how you will locate point
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.
You have “constructed” a perpendicular bisector of
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
Now that you have created a line perpendicular to
Label the midpoint of
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.
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
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.
Use a straightedge to draw the two diagonals.
Using a compass, construct a circle with center at point
and a radius of length . Then construct a circle at point with radius length .
What do you notice about the intersections of the two circles?
Solve the system of equations. Choose an appropriate method: graphing, substitution, or elimination.