# Lesson 7: Building Polygons (Part 2)

Let’s build more triangles.

## 7.1: Where Is Lin?

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

1. Draw a diagram of the situation including a place where Lin could be.

2. How far away from the swings is Lin in your diagram?

3. Where are some other places Lin could be?

GeoGebra Applet fhm2Tkk9

## 7.2: How Long Is the Third Side?

Use the applet to answer the questions.

1. Build as many different triangles as you can that have one side length of 5 inches and one of 4 inches. Record the side lengths of each triangle you build.

2. Are there lengths that could be used for the third side of the triangle that aren't values of the sliders?

3. Are there lengths that are values of the sliders that could not be used as the third side of the triangle?

GeoGebra Applet yrZKQ9r8

## 7.3: Swinging the Sides Around

Use the applet to answer the questions.

1. If you wanted to make a triangle with side lengths of 3 units, 3 units, and 4 units, how would you know at what angle you should position the sides?

2. Let segment $AC$ be the 3-unit side length. Right-click on point $C$, check Trace On, and drag point $C$.

3. What path does point $C$ trace as you move it around? Why? What other tool do we have in our geometry toolkit that could do something similar?

4. Using 2 different colors, draw 2 different triangles that would have the 4-unit segment as their base and a 3-unit segment for their right side.

5. Right-click on point $C$, uncheck Trace On.

6. Repeat the previous instructions, letting segment $BD$ be the 3-unit side length.

7. Using a third color, draw a point where the two traces intersect. Next, draw another triangle that would have segment $AB$ as its base and segments $AC$ and $BD$ for both of its other sides.

8. If you had used segments that were only 2 units long on either end of segment $AB$, how would that have affected your drawing?

GeoGebra Applet PHQxsCFp

## 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:

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

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:

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.

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.