Lesson 15Building Polygons

Learning Goal

Let’s build shapes.

Learning Targets

I can show that the 3 side lengths that form a triangle cannot be rearranged to form a different triangle.
I can show that the 4 side lengths that form a quadrilateral can be rearranged to form different quadrilaterals.
I can show whether or not 3 side lengths will make a triangle.

Warm Up: Where Is Lin?

At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide.

open applet in presentation mode

Draw a diagram of the situation including a place where Lin could be.
How far away from the swings is Lin in your diagram?
Where are some other places Lin could be?

Print Version

At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide.

Draw a diagram of the situation including a place where Lin could be.
How far away from the swings is Lin in your diagram?
Where are some other places Lin could be?

Activity 1: Building Diego’s and Jada’s Shapes

Problem 1

Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.

open applet in presentation mode

Build such a shape.
Is your shape an identical copy of Diego’s shape? Explain your reasoning.

Print Version

Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.

Build such a shape.
Is your shape an identical copy of Diego’s shape? Explain your reasoning.

Problem 2

Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.

open applet in presentation mode

Build such a shape.
Is your shape an identical copy of Jada’s shape? Explain your reasoning.

Print Version

Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.

Build such a shape.
Is your shape an identical copy of Jada’s shape? Explain your reasoning.

Activity 2: Swinging the Sides Around

We’ll explore a method for drawing a triangle that has three specific side lengths. Use the applet to answer the questions.

open applet in presentation mode

Follow these instructions to mark the possible endpoints of one side:
- For now, ignore segment $A C$ , the 3-inch side length on the left side
- Let segment $B D$ be the 3-unit side length on the right side. Right-click on point $D$ , check Trace On. Rotate the point, drawing all the places where a 3-inch side could end.
What shape have you drawn while moving $B D$ around? Why? Which tool in your geometry toolkit can do something similiar?
Use your drawing to create two unique triangles, each with a base of length 4 inches and a side of length 3 inches. Use a different color to draw each triangle.
Repeat the previous instructions, letting segment $A C$ be the 3-unit side length.
Using a third color, draw a point where the two traces intersect. Using this third color, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches.

Print Version

We’ll explore a method for drawing a triangle that has three specific side lengths. Your teacher will give you a piece of paper showing a 4-inch segment as well as some instructions for which strips to use and how to connect them.

Follow these instructions to mark the possible endpoints of one side:
- Put your 4-inch strip directly on top of the 4-inch segment on the piece of paper. Hold it in place.
- For now, ignore the 3-inch strip on the left side. Rotate it so that it is out of the way.
- In the 3-inch strip on the right side, put the tip of your pencil in the hole on the end that is not connected to anything. Use the pencil to move the strip around its hinge, drawing all the places where a 3-inch side could end.
- Remove the connected strips from your paper.
What shape have you drawn while moving the 3-inch strip around? Why? Which tool in your geometry toolkit that can do something similar?
Use your drawing to create two unique triangles, each with a base of length 4 inches and a side of length 3 inches. Use a different color to draw each triangle.
Reposition the strips on the paper so that the 4-inch strip is on top of the 4-inch segment again. In the 3-inch strip on the left side, put the tip of your pencil in the hole on the end that is not connected to anything. Use the pencil to move the strip around its hinge, drawing all the places where another 3-inch side could end.
Using a third color, draw a point where the two marks intersect. Using this third color, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches.

Lesson Summary

If we want to build a polygon with two given side lengths that share a vertex, we can think of them as being connected by a hinge that can be opened or closed:

A point with six lines extending from it. The first is length 4 and the remaining lines are 3.

All of the possible positions of the endpoint of the moving side form a circle:

A circle with 5 radius noted at 3 and a line extending outside the circle with length 4.

You may have noticed that sometimes it is not possible to build a polygon given a set of lengths. For example, if we have one really, really long segment and a bunch of short segments, we may not be able to connect them all up. Here’s what happens if you try to make a triangle with side lengths 21, 4, and 2:

A partial triangle with side lengths 4, 21, and 2.

The short sides don’t seem like they can meet up because they are too far away from each other.

If we draw circles of radius 4 and 2 on the endpoints of the side of length 21 to represent positions for the shorter sides, we can see that there are no places for the short sides that would allow them to meet up and form a triangle.

A partial triangle with side lengths 4, 21, and 2. Side lengths 4 and 2 and shown as the radius within a circle.

In general, the longest side length must be less than the sum of the other two side lengths. If not, we can’t make a triangle!

If we can make a triangle with three given side lengths, it turns out that the measures of the corresponding angles will always be the same. For example, if two triangles have side lengths 3, 4, and 5, they will have the same corresponding angle measures.