## Learning Math: Geometry

# 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.

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.

### Solutions

**Problem C1**

Start by drawing a line segment. Then do the following:

**a.** Fold the paper so that one of the endpoints of the line segment lies somewhere on the line segment. The crease created defines a line perpendicular to the original line segment.

**b. **Use the process above to construct a perpendicular line. Then use the same process to construct a line perpendicular to the new line, making sure that this second perpendicular is a different line from the original. Since this third line and the original are each perpendicular to the second line, they are parallel.

**c.** Fold the paper so that the endpoints of the line segment overlap. Draw a line segment along the crease, intersecting the original line segment. This new line segment is perpendicular to the original one and bisects it, because we used the same process that we used to construct the midpoint in the sample construction.

**Problem C2**

Draw an angle on a piece of paper. Next, fold the paper so that the two sides of the angle overlap. The crease created defines a bisector of the angle.

**Problem C3**

For parts (a)-(c), draw several triangles, at least one of which has an obtuse angle (to see that the definitions make sense in general). Then draw in the altitudes. Repeat with medians. Repeat with midlines.

**a.** altitudes:

**b.** medians:

**c. **midlines:

**Problem C4**

Draw five triangles on separate pieces of patty paper, and then do the following:

**a. **Pick a side. Fold the paper so that the crease is perpendicular to the side [see Problem C1(a)] and so that it goes through the vertex opposite the side. You may have to extend the line segments of your triangle if the triangle has an angle larger than 90°. (See illustration for an example of what this looks like.) Connect the side with the vertex along the crease. The line segment drawn is the altitude corresponding to the side chosen. Now repeat with the other two sides.

**b.** Pick a side. Fold the paper so that the endpoints of the chosen side overlap. The midpoint of the side is the point where the side intersects the crease. Using a straightedge, connect the midpoint of the side with the vertex opposite it. Repeat with the other two sides.

**c.** Pick a side. Find the midpoint of the side by following the construction of question (b). Repeat this construction with the other two sides. Using a straightedge, connect the consecutive midpoints.

**d.** Pick a side. Construct a perpendicular bisector of the chosen side using the construction from Problem C1(c). Repeat with the other two sides.

**e.** Pick an angle. Fold the paper so that the two sides of this angle overlap. The crease defines a ray that bisects the chosen angle. Repeat with the other two angles.

**Problem C5**

Draw a line segment, and then do the following:

**a. **Make a crease that goes through one of the endpoints of the original line segment. The crease will extend to the edges of the paper.

Fold the paper along the crease and mark where the second endpoint of the line segment overlaps with the folded piece.

Then unfold the paper and connect the marked point with the point where the original line segment and the crease intersect.

Next, connect the two “free” ends of the two line segments with a straight line. The result is an isosceles triangle, where the original line segment is one of its two equal sides.

**b. **Construct a perpendicular bisector of the line segment. Choose any point on the perpendicular bisector and connect it with the endpoints of the original line segment. The resulting triangle is isosceles and has the original line segment as its base.

**c.** Extend the line segment to form a line, being sure to mark the original endpoints of the line segment. Use this line to construct a perpendicular line through the endpoints of the original line segment [see Problem C1(a)].

You should now have two parallel lines, each perpendicular to the original line segment at the endpoints.

Then fold the paper so that the original line segment overlaps the first perpendicular line you drew.

Mark the point on the perpendicular line where the second endpoint falls on this line. (This defines one of the equal, perpendicular sides.)

Perform the same process on the second perpendicular line to define the third side of the square.

Finally, use the straightedge to connect the two points you marked on the perpendicular lines to define the fourth side of the square.

**d. **Construct the perpendicular bisector of the segment.

Pick one of the two endpoints of the original line segment. Fold the paper along a line that contains the endpoint such that the other endpoint falls on the perpendicular bisector.

Flip your patty paper, and then mark that spot on the bisector.

Connect the marked spot with the two endpoints of the original line segment.

**Problem C6**

**a.** Fold the paper in half to make a rectangle. Fold it in half again by bisecting the longer sides of the rectangle. The resulting square has one-fourth the area of the original one. There are exactly four squares that fit exactly on top of each other, so they must have the same area. Since together they completely make up the original square, each must be one-fourth of the original square.

**b.** Find the midpoints of all four sides. Connect the consecutive midpoints. The resulting square has one-half the area of the original square. To see this, connect the diagonals of the new square. You will see four triangles inside the square and four triangles outside, all of which have the same area. Half the area of the original square is inside the new square.

**c.** In order to obtain a square with exactly three-fourths the area of the original square (sides = 1) we need to calculate the sides of the new square:

a • a = 3/4

a2 = 3/4

a = √3/2

So we are looking to construct a square whose sides are equal to √3/2.

Start with a piece of patty paper. Think of the bottom edge as the base of an equilateral triangle.

Fold the vertical midline. The third vertex of the equilateral triangle will be on this midline.

Bring the lower right vertex up to the midline, so that the entire length of the bottom edge is copied from the lower left vertex to the midline.

The distance from the bottom of the midline to this mark is √3/2. It is the height of an equilateral triangle with the side length equal to 1. This can also be easily calculated using the Pythagorean theorem for the right triangle with the sides 1 and 1/2.

Next, fold down from this mark. This is the height of the square.

Fold down the corner so that you can fold the right side in the same amount as the top.

Fold over the right side. This square has area 3/4 of the original square.

**Problem C7**

One way to do this is to use a straightedge to draw the two diagonals of the square. The centroid is the point of their intersection. Another is to draw the perpendicular bisectors of two consecutive sides of the square [the same construction as Problem C6(a)]. The intersection of these bisectors is the same centroid.

**Problem C8**

Noticing what appear as concurrencies in the folds may lead one to conjecture that concurrencies occur in general. Keep in mind any one construction that suggests this is a special case. Therefore, in order to convince ourselves that they do occur in general, we need to construct a formal proof.