Private: Learning Math: Geometry

Professional Development > Private: Learning Math: Geometry > 1. What Is Geometry? > 1.3 Part C: Folding Paper (40 minutes)

Mathematics

K-2, 3-5, 6-8

What Is Geometry? Part C: Folding Paper (40 minutes)

Session 1, Part C

In This Part:

Constructions
Constructing Triangles
Concurrencies in Triangles
More Constructions

Geometers distinguish between a drawing and a construction. Drawings are intended to aid memory, thinking, or communication, and they needn’t be much more than rough sketches to serve this purpose quite well. The essential element of a construction is that it is a kind of guaranteed recipe. It shows how a figure can be accurately drawn with a specified set of tools. A construction is a method, while a picture merely illustrates the method.

The most common tools for constructions in geometry are a straightedge (a ruler without any markings on it) and a compass (used for drawing circles). In the problems below, your tools will be a straightedge and patty paper. You can fold the patty paper to create creases. Since you can see through the paper, you can use the folds to create geometric objects. Though your “straightedge” might actually be a ruler, don’t measure! Use it only to draw straight segments. See Note 4 below.

Throughout this part of the session, use just a pen or pencil, your straightedge, and patty paper to complete the constructions described in the problems. Here is a sample construction with patty paper to get you started:

To construct the midpoint of a line segment, start by drawing a line segment on the patty paper.

Next, fold the paper so that the endpoints of the line segment overlap. This creates a crease in the paper.

The intersection of the crease and the original line segment is the midpoint of the line segment.

Constructions

To construct a perpendicular line, consider that a straight line is a 180° angle. Can you cut that angle in half (since perpendicular lines form right angles, or 90° angles)? To construct a parallel line, you may need to construct another line before the parallel to help you.

Problem C1

Draw a line segment. Then construct a line that is

perpendicular to it
parallel to it
the perpendicular bisector of the segment (A perpendicular bisector is perpendicular to the segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.)

Problem C2

Draw an angle on your paper. Construct its bisector. (An angle bisector is a ray that cuts the angle exactly in half, making two equal angles.)

Constructing Triangles

Problem C3

Illustrate each of these definitions with a sketch using four different triangles. Try to draw four triangles that are different in significant ways — different side lengths, angles, and types of triangles. The first one in definition (a) is done as an example.

A triangle has three altitudes, one from each vertex. (An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.)
A triangle has three medians. (A median is a segment connecting any vertex to the midpoint of the opposite side.)
A triangle has three midlines. (A midline connects two consecutive midpoints.)

When you construct medians, you need to do two things: First find the midpoint; then fold or draw a segment connecting that point to the opposite vertex. Except in the case of special triangles (such as an equilateral triangle, and one median in an isosceles triangle), you can’t construct a median with just one fold. When you construct altitudes, you need to construct a perpendicular to a segment, but not necessarily at the midpoint of that segment. And remember that the altitude may fall outside the triangle, so you might want to draw or fold an extension of the sides of the triangle to help you.

Problem C4

Draw five triangles, each on its own piece of patty paper. Use one triangle for each construction below.

Carefully construct the three altitudes of the first triangle.
Carefully construct the three medians of the second triangle.
Carefully construct the three midlines of the third triangle.
Carefully construct the three perpendicular bisectors of the fourth triangle.
Carefully construct the three angle bisectors of the fifth triangle.

VIDEO SEGMENT

In this video segment, participants construct the altitudes, medians, and midlines of their triangles. Compare your solutions to Problem C4 with those in this video segment.

What are the similarities and differences in your results? What conjectures can you make about the constructions you’ve just completed?

You can find this segment on the session video approximately 15 minutes and 40 seconds after the Annenberg Media logo.

Problems C3 and C4 and the Video Segment problems taken from Connected Geometry, developed by Educational Development Center, Inc. p. 32. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

Concurrencies in Triangles

When three or more lines meet at a single point, they are said to be concurrent. The following surprising facts are true for every triangle:

The medians are concurrent; they meet at a point called the centroid of the triangle. (This point is the center of mass for the triangle. If you cut a triangle out of a piece of paper and put your pencil point at the centroid, you would be able to balance the triangle there.)

The perpendicular bisectors are concurrent; they meet at the circumcenter of the triangle. (This point is the same distance from each of the three vertices of the triangles.)

The angle bisectors are concurrent; they meet at the incenter of the triangle. (This point is the same distance from each of the three sides of the triangles.)

The altitudes are concurrent; they meet at the orthocenter of the triangle.

Triangles are the only figures where these concurrencies always hold. (They may hold for special polygons, but not for just any polygon of more than three sides.) We’ll revisit these points in a later session and look at some explanations for why some of these lines are concurrent. You’ll explore the derivation of such terms as incenter and circumcenter later in Session 5 of this course.

TAKE IT FURTHER

Problem C5

For each construction in parts a-d, start with a freshly drawn segment on a clean piece of patty paper. Then construct the following shapes:

an isosceles triangle with your segment as one of the two equal sides
an isosceles triangle whose base is your segment
a square based on your segment
an equilateral triangle based on your segment

More Constructions

Think about how you might construct the exact side lengths needed for these squares. For example, the first square will need a side length exactly one half the original. The third square is very difficult!

Problem C6

Start with a square sheet of paper.

Construct a square with exactly one-fourth the area of your original square. How do you know that the new square has one-fourth the area of the original square?
Construct a square with exactly one half the area of your original square. How do you know that the new square has one half the area of the original square?
Construct a square with exactly three-fourths the area of your original square.

Problem C7

Recall that the centroid is the center of mass of a geometric figure. How could you construct the centroid of a square?

Take It Further

Problem C8

When you noticed concurrencies in the folds, were you sure that the segments were concurrent? What would convince you that, for example, the medians of every triangle really are concurrent?

Notes

Note 4

If you are working in a group, you may choose to do all of the construction problems as a group activity. Watch for someone with appropriate solutions (for example, folding the two endpoints to each other, rather than measuring). Ask that person to share the solution and explain why it will always work on any segment. That’s the goal for these problems: to come up with general methods that will always work and that don’t rely on measurement. At the end, leave at least 10 minutes to share methods, even if not everyone is done. Then make a list of conjectures that come from that problem.