7.2.2: Building Polygons (Part 2)

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Illustrative Mathematics
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Lesson

Let's build more triangles.

Exercise $\PageIndex{1}$: Where is Lin?

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

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?

Exercise $\PageIndex{2}$: How Long is the Third Side?

Use the applet to answer the questions.

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.
Are there any other lengths that could be used for the third side of the triangle but aren't values of the sliders?
Are there any lengths that are values of the sliders but could not be used as the third side of the triangle?

Are you ready for more?

Assuming you had access to strips of any length, and you used the 9-inch and 5-inch strips as the first two sides, complete the sentences:

The third side can't be _____ inches or longer.
The third side can't be _____ inches or shorter.

Exercise $\PageIndex{3}$: 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.

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
  $clipboard_e5407b601d4fb60228b7512eccc421fb8.png$
  Figure $\PageIndex{1}$
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.
What shape have you drawn while moving $BD$ around? Why? Which tool in your geometry toolkit 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.
Repeat the previous instructions, letting segment $AC$ 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.

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.

Practice

Exercise $\PageIndex{4}$

In the diagram, the length of segment $AB$ is 10 units and the radius of the circle centered at $A$ is 4 units. Use this to create two unique triangles, each with a side of length 10 and a side of length 4. Label the sides that have length 10 and 4.

Exercise $\PageIndex{5}$

Select all the sets of three side lengths that will make a triangle.

$3, 4, 8$
$7, 6, 12$
$5, 11, 13$
$4, 6, 12$
$4, 6, 10$

Exercise $\PageIndex{6}$

Based on signal strength, a person knows their lost phone is exactly 47 feet from the nearest cell tower. The person is currently standing 23 feet from the same cell tower. What is the closest the phone could be to the person? What is the furthest their phone could be from them?

Exercise $\PageIndex{7}$

Each row contains the degree measures of two complementary angles. Complete the table.

Table $\PageIndex{1}$
measure of an angle	measure of its complement
$80^{\circ}$
$25^{\circ}$
$54^{\circ}$
$x$

(From Unit 7.1.2)

Exercise $\PageIndex{8}$

Here are two patterns made using identical rhombuses. Without using a protractor, determine the value of $a$ and $b$. Explain or show your reasoning.

(From Unit 7.1.1)

Exercise $\PageIndex{9}$

Mai’s family is traveling in a car at a constant speed of 65 miles per hour.

At that speed, how long will it take them to travel 200 miles?
How far do they travel in 25 minutes?

(From Unit 4.1.3)

measure of an angle	measure of its complement
\(80^{\circ}\)
\(25^{\circ}\)
\(54^{\circ}\)
\(x\)

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