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5.5 – What Patterns Can I Use?

Constant Ratios in Right Triangles

So far in this chapter you have learned how to find the sides of special right triangles. But what if the triangle isn’t special? Today we are going to focus our attention on slope triangles, which were used in algebra to describe linear change.

5.24 – PATTERNS IN SLOPE TRIANGLES

Today you are going to focus on the relationship between the angles and the sides of a right triangle. You will start by studying slope triangles. Notice in the graph shown, a slope triangle is drawn for you.

a. Draw two new slope triangles on the line. Each should be a different size. Label each triangle with as much information as you can, such as its horizontal and vertical lengths and its angle measures.

2

10

1

5

2

10

=

a. Draw two new slope triangles on the line. Each should be a different size. Label each triangle with as much information as you can, such as its horizontal and vertical lengths and its angle measures.

3

15

3

15

1

5

=

a. Draw two new slope triangles on the line. Each should be a different size. Label each triangle with as much information as you can, such as its horizontal and vertical lengths and its angle measures.

4

20

4

20

1

5

=

b. What do these triangles have in common? How are these triangles related to each other?

similar by AA~

d. What do you notice about the slope ratios written in fraction form? What do you notice about the decimals?

The ratios are equal

e. Notice how the ∆ fraction form? What do you notice about the decimals?y is on the opposite side of triangle from where the angle is. What side is the ∆x? How is the adjacent side different from the hypotenuse?

The rise is opposite the angle

The run is adjacent to the angle, but not the hypotenuse

5.25 – PROPORTIONS fraction form? What do you notice about the decimals?

Tara thinks she sees a pattern in these slope triangles, so she decides to make some changes in order to investigate whether or not the patterns remain true.

a. She asks, "What if I drew a slope triangle on this line with ∆y = 6? What would the ∆x be for that slope triangle? Answer her question and explain how you figured it out.

b. "What if ∆ fraction form? What do you notice about the decimals?x = 40?" she wonders. "Then what is ∆y?" Find ∆y, and explain your reasoning.

c. Tara wonders, "What if I draw a slope triangle on a different line? Can I still use the same ratio to find a missing ∆x or ∆y value?" Discuss this with your team and explain to Tara what she could expect.

No, the triangles won’t be similar

5.26 – CHANGING LINES different line? Can I still use the same ratio to find a missing ∆

In part (c) of the last problem, Tara asked, "What if I draw my triangle on a different line?" With your team, investigate what happens to the slope ratio and slope angle when the line is different. Use the graph grids below to graph the lines described. Use the graphs and your answers to the questions below to respond to Tara's question.

c. Graph the line different line? Can I still use the same ratio to find a missing ∆y = x + 4 on the graph. Draw a slope triangle and label its horizontal and vertical lengths. What is the new slope ratio? What is the slope angle?

1

m =

3

45°

3

5.27 – TESTING CONJECTURES different line? Can I still use the same ratio to find a missing ∆

The students in Ms. Matthews class are writing conjectures based on their work today. As a team, decide if you agree or disagree with each of the conjectures below. Explain your reasoning.

False different line? Can I still use the same ratio to find a missing ∆

True

True

False, lines can be parallel with same slope

5.29 – ANOTHER LOOK different line? Can I still use the same ratio to find a missing ∆

Sheila says that the triangle in part (f) of the previous problem is the same as the picture below.

a. Do you agree? Why or why not?

Yes, AA~

b. Use what you know about the slope ratio of 11 different line? Can I still use the same ratio to find a missing ∆° to find the slope ratio for 79°.

5

1

c. What is the relationship of 11 different line? Can I still use the same ratio to find a missing ∆° and 79°? Of their slope ratios?

Angles are complementary

Ratios are reciprocal

5.30 – MAKING CONNECTIONS different line? Can I still use the same ratio to find a missing ∆

For what other angles can you find the slope ratios based on your work?

a. For example, since you know the slope ratio of 22°, what other angle do you know the slope ratio for? Find the complement of each slope angle you know.

52

31

11

68° is

72° is

45° is

b. Use this information to find different line? Can I still use the same ratio to find a missing ∆x in the diagram at right.

c. Complete the conjecture about the relationship of the slope ratios for complementary angles.

b

a

complementary

possible slope ratios for complementary angles.

possible slope ratios for complementary angles.

possible slope ratios for complementary angles.

possible slope ratios for complementary angles.

Triangle is slope ratios for complementary angles.not possible, but degree is

possible slope ratios for complementary angles.

Triangle is slope ratios for complementary angles.not possible

Degree is not possible

5.32 – COMPLETING THE CHART slope ratios for complementary angles.

a. What happens to the slope ratio when the angle increases? Decreases?

increases

decreases

573

10

= 57.3

- When is a slope ratio more than 1? 90°?
- When is it less than 1?
- When is it equal to 1?

573

10

= 57.3

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