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Page 1: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Triangles

Page 2: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

9.2

The Pythagorean Theorem

Page 3: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

The Pythagorean Theorem

In a right triangle, the sum of the legs squared equals the hypotenuse squared.

a2 + b2 = c2, where a and b are legs and c is the hypotenuse.

ac

b

Page 4: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Pythagorean Triples

Pythagorean TripleWhen the sides of a right triangle are all

integers it is called a Pythagorean triple.

3,4,5 make up a Pythagorean triple since 32 + 42 = 52.

Page 5: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 1

Find the unknown side lengths. Determine if the sides form a Pythagorean triple.

6

8

48

50yx

Page 6: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 2

Find the unknown side lengths. Determine if the sides form a Pythagorean triple.

100

q 90

90

50p

Page 7: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 3

Find the unknown side lengths. Determine if the sides form a Pythagorean triple.

2

3

e

1715

d

Page 8: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 4

Find the unknown side lengths. Determine if the sides form a Pythagorean triple.

4 3

5

g

8f

5 3

Page 9: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

9.3

The Converse of the Pythagorean Theorem

Page 10: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

a

c 2 = a 2 + b 2

b

a

b

If a and b stay the same length and we make the angle between them smaller, what happens to c?

Page 11: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

If a and b stay the same length and we make the angle between them bigger, what happens to c?

a

c 2 = a 2 + b 2

b

a

b

Page 12: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Classifying Triangles

Let c be the biggest side of a triangle, and a and b be the other two side.

If c2 = a2 + b2, then the triangle is right. If c2 < a2 + b2, then the triangle is acute. If c2 > a2 + b2, then the triangle is obtuse.

*** If a + b is not greater than c, a triangle cannot be formed.

Page 13: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 1

Determine what type of triangle, if any, can be made from the given side lengths.

7, 8, 12

11, 5, 9

Page 14: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 2

Determine what type of triangle, if any, can be made from the given side lengths.

5, 5, 5

1, 2, 3

Page 15: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 3

Determine what type of triangle, if any, can be made from the given side lengths.

16, 34, 30

9, 12, 15

Page 16: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 4

Determine what type of triangle, if any, can be made from the given side lengths.

13, 5, 7

13, 18, 22

Page 17: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 5

Determine what type of triangle, if any, can be made from the given side lengths.

4, 8,

5, , 5

4 3

5 2

Page 18: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

9.4

Special Right Triangles

Page 19: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

45º-45º-90º Triangles

Solve for each missing side. What pattern, if any do you notice?

2

2

3

3

Page 20: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

45º-45º-90º Triangles

4

4

5

5

Page 21: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

45º-45º-90º Triangles

6

6

7

7

Page 22: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

45º-45º-90º Triangles

300

300

½

½

Page 23: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

45º-45º-90º Triangles

x

x

Page 24: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

45º-45º-90º Triangles

In a 45º-45º-90º triangle, the hypotenuse is times each leg.

2

x

x2x

Page 25: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

30º-60º-90º Triangles

Solve for each missing length. What pattern, if any do you notice?

10 10

10

Page 26: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

30º-60º-90º Triangles

8 8

8

Page 27: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

30º-60º-90º Triangles

6 6

6

Page 28: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

30º-60º-90º Triangles

50 50

50

Page 29: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

30º-60º-90º Triangles

2x 2x

2x

Page 30: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

30º-60º-90º Triangles

In a 30º-60º-90º triangle, the hypotenuse is twice as long as the shortest leg, and the longer leg is times as long as the shorter leg. 3

2x

x

30º

60º

3x

Page 31: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 1

Find each missing side length.

45º

15 45º

6

Page 32: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 2

45º

12

30º

18

Page 33: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 3

30º

12

30º

44

Page 34: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

2x

x

30º

60º

3x

x

x2x

Page 35: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

9.5

Trigonometric Ratios

Page 36: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Warm Up

Name the side opposite angle A. Name the side adjacent to angle A. Name the hypotenuse.

A

C B

Page 37: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Trigonometric Ratios

The 3 basic trig functions and their abbreviations aresine = sincosine = cos tangent = tan

Page 38: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

SOH CAH TOA

sin = opposite side hypotenuse

cos = adjacent sidehypotenuse

tan = opposite sideadjacent side

SOH

CAH

TOA

Page 39: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 1

Find each trigonometric ratio.sin Acos A tan Asin Bcos B tan B

3

4C

A

5

B

Page 40: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 2

Find the sine, the cosine, and the tangent of the acute angles of the triangle. Express each value as a decimal rounded to four decimal places.

7

24E

D

25

F

Page 41: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

9.6

Solving Right Triangles

Page 42: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 1

Find the value of each variable. Round decimals to the nearest tenth.

25º

8a

Page 43: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 2

42º

40

b

Page 44: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 3

20º

c

8

Page 45: Triangles. 9.2 The Pythagorean Theorem In a right triangle, the sum of the legs squared equals the hypotenuse squared. a 2 + b 2 = c 2, where a and b

Example 4

17º

10

c


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