In Chapter 4, you investigated similarity and discovered that similar triangles have special relationships. In this chapter, you will discover that the side ratios in a right triangle can serve as a powerful mathematical tool that allows you to find missing side lengths and missing angle measures for any right triangle. You will also learn how these ratios (called trigonometric ratios) can be used in solving problems.

5.1

What If The Triangles Are Special?

Pg. 3Isosceles Right Triangles and Squares

5.3 – What If The Triangles Are Special?Isosceles Right Triangles and Squares

You now know when triangles are similar and how to find missing side lengths in a similar triangle using proportions. Today you will be using both of these ideas to investigate patterns within special isosceles right triangles. These patterns will allow you to use a shortcut whenever you are finding side lengths in these particular types of right triangles.

5.1 – SQUARESa. Find the length of a diagonal in a square with a side length of 1cm.

1cm

1cm

2 2 21 1 x 21 1 x 22 x

2 x

2

b. Find the length of a diagonal in a square with a side length of 5cm.

5cm

5cm

2 2 25 5 x 225 25 x

250 x

5 2 x

5 2

c. Find the length of a diagonal in a square with a side length of 7cm.

7cm

7cm

2 2 27 7 x 249 49 x

298 x

7 2 x

7 2

d. Since all squares are similar, Jebari decided to follow the pattern to find the missing diagonal length using ratios. Without using the Pythagorean theorem, use the pattern to find all of the missing lengths of the square.

25

25

2525 2

30

30 3030 2

8 8

8

8

10

22

2 10 2

25 2

5 2

5 2

5 2

5 2

d. Jebari noticed that when you draw the diagonal of a square it makes two isosceles triangles. Given this fact, find all of the missing angles in the given picture. Then find the missing sides in respect to x.

x

2x

x

x

4545

4545

e. What if it is only a half square? Find the missing sides of the isosceles right triangle. Then complete the relation.

If the side opposite the 45° is ______, the

side opposite the other 45° is ______, and

opposite the 90° is ______.

9

x 2

x

x

x

x 2

9 2

45

8

8 2

4511

11 2

9

945

15 2

15

45

5545

14 2

1445

45

8

22

2 8 2

2 4 2

4 2

4 2

12

22

2 12 2

26 2

6 2

6 245

5.3 – DIAGONALS OF OTHER SHAPESEva was very excited to find this shortcut. She decided to use this to find the length of a diagonal for a rectangle.

a. What was wrong with what Eva wants to do? Explain why you can't use 45°-45°-90° with a rectangle. 

The two legs of the triangle are not equal

b. What should she do instead of using 45°-45°-90° ratios?

Pythagorean theorem

c. Find the length of a diagonal in the given rectangle.

2 2 25 9 x 225 81 x 2106 x

106 x

106

5.4 – AREA Find the area of the shapes using the special triangle ratio.

10 14

1414

2

bhA bh

14 14

2A 14 10

98A 140

2238A un

6

6

6 6A

236A un

10 2

10 2

10 2 10 2A

2200A un

Right Triangles Project

Take 5 computer papers and fold them to make a flip chart. Staple the top of the papers.

Right Triangles Project

Pythagorean Theorem: Given 2 sides

45º– 45º– 90º

30º– 60º– 90º

Sine – S

sin-1, cos-1, tan-1

Your NameBlock#

Cosine – C

Tangent – T

2x x x 3 2x x x

OH

AHOA

Clinometer MeasuresArea of Regular Polygons

Pythagorean Theorem: Given two side lengths

leg

leghypotenuse

leg2 + leg2 = hyp2

5 11

x

45º– 45º– 90º x x x 2

8

45°

45°

y x

x

45°

45°

y 12

45 45 90 1 1 2

x

x

45°

45°

2x