# Lesson 15

Building Polygons

Let’s build shapes.

### 15.1: 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?

### 15.2: Building Diego’s and Jada’s Shapes

1. Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.
1. Build such a shape.
2. Is your shape an identical copy of Diego’s shape? Explain your reasoning.
2. Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.
1. Build such a shape.

### 15.3: 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.

1. Follow these instructions to mark the possible endpoints of one side:

1. For now, ignore segment $$AC$$ , the 3-inch side length on the left side

2. Let segment $$BD$$ 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.

2. What shape have you drawn while moving $$BD$$ around?  Why?  Which tool in your geometry toolkit can do something similiar?

3. 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.

4. Repeat the previous instructions, letting segment $$AC$$ be the 3-unit side length.

5. 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.

### 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.