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www.njctl.org 2014-06-05

Geometry Trigonometry of Right Triangles

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Table of Contents Pythagorean Theorem Special Right Triangles Trigonometric Ratios Solving Right Triangles Angles of Elevation and Depression Area of an Oblique Triangle

Click on a Topic to go to that section

Similarity in Right Triangles Law of Sines and Law of Cosines

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Pythagorean Theorem

Return to the Table of Contents

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Before learning about similar right triangles and trigonometry, we need to review the Pythagorean Theorem and the Pythagorean Theorem Converse.

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Recall that a right triangle is a triangle with a right angle. The sides form that right angle are the legs. The side opposite the right angle is the hypotenuse. The hypotenuse is also the longest side.

leg hypotenuse leg

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In a right triangle, the sum of the squares of the lengths

• f the legs is equal to the square of the length of the

hypotenuse. leg2 + leg2 = hypotenuse2

• r

a2 + b2 = c2 a b c

Pythagorean Theorem Slide 7 / 240

Example: Find the length of the missing side of the right triangle.

x 9 12

Is the missing side a leg or the hypotenuse of the right triangle? Answer

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92 + 122 = x2 81 + 144 = x2 225 = x2 15 = x

• 15 is a extraneous solution, a distance

can not equal a negative number. x = 15



x

9

12

Solve for x:

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x 28 20

Is the missing side a leg or the hypotenuse

• f the right triangle?

Example: Find the length of the missing side

• f the right triangle.

Answer

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1 The missing side is the ________ of the right triangle.

A leg B

hypotenuse

6 9 x

Answer

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2 Find the length of the missing side.

6 9 x

Answer

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3 The missing side is the _________ of the right triangle.

A leg B

hypotenuse

x 15 36

Answer

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4 Find the length of the missing side.

x 15 36

Answer

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The safe distance of the base of the ladder from a wall it leans against should be one-fourth of the length of the ladder.

28 feet 7 feet

?

Thus, the bottom of a 28-foot ladder should be 7 feet from the wall. How far up the wall will a ladder reach?

Real World Application Slide 15 / 240

28 feet 7 feet

?

The ladder will reach feet up the wall safely. Answer Solve using a + b = c

2 2 2

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84 50

x

The dimensions of a high school basketball court are 84' long and 50' wide. What is the length from one corner of the court to the opposite corner?

Real World Application

Answer

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5 A NBA court is 50 feet wide and the length from

• ne corner of the court to the opposite corner is

106.5 feet. How long is the court?

A 94.03 feet B

117.7 feet

C

118 feet

D

94 feet

(Round the answer to the nearest whole number) Answer

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Pythagorean Theorem Applications

The Pythagorean Theorem can also be used in figures that contain right angles.

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Example Find the perimeter of the square. note: Before finding the perimeter of the square, we need to first find the length

• f each side.

18 cm

Psq = 4s

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18 cm

x Remember, in a square all sides are congruent. Answer Start here: x + x = 18

2 2 2

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Example Find the area of the triangle. The base of the triangle is given, but we need to find the height of the triangle. A = bh

1 2

13 feet 10 feet 13 feet

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By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base.

13 feet 5 feet 13 feet

h

5 feet

Answer

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Try this... Find the perimeter of the rectangle.

Prect = 2l + 2w

8 in 10 in

Answer

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6 Find the area of the rectangle.

A 120 square feet B 84 square feet C

46 square inches

D

46 square feet

8 feet 1 7 f e e t

Answer

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7 Find the perimeter of the square. (Round to the nearest tenth)

A 12.8 cm B

25.5 cm

C

25.6 cm

D

36 cm

9 c m

Answer

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8 Find the area of the triangle.

7 inches 10 inches 7 inches

Answer

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9 Find the area of the triangle.

7 inches 4 inches 7 inches

Answer

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If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, then ABC is a right triangle.

a b c

A B C

Converse of the Pythagorean Theorem Slide 29 / 240

Example Tell whether the triangle is a right triangle.

Remember c is the longest side

D E F 7 24 25

Answer

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If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. If c2 > a2 + b2, then ABC is obtuse.

A B C a b c

Theorem Slide 31 / 240

If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. If c2 < a2 + b2 , then ABC is acute.

a b c A B C

Theorem Slide 32 / 240

Example Classify the triangle as acute, right, or obtuse.

17 15 13

Answer

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10 Classify the triangle as acute, right, obtuse, or not a triangle.

A acute B

right

C

• btuse

D

not a triangle 11 12 15

Answer

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11 Classify the triangle as acute, right, obtuse, or not a triangle.

A acute B

right

C

• btuse

D

not a triangle

6 3 5

Answer

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12 Classify the triangle as acute, right, obtuse, or not a triangle.

A acute B

right

C

• btuse

D

not a triangle

25 19 20

Answer

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13 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle.

A acute B

right

C

• btuse

Answer

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14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle.

A acute triangle B

right triangle

C

• btuse triangle

Answer

Slide 38 / 240 Review

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

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Similarity in Right Triangles

Return to the Table of Contents

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There are many proofs to the Pythagorean Theorem. How many do you know? Triangle similarity can be used to prove the Pythagorean Theorem. How?

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Theorem The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. CD is the altitude of ABC

ABC~ ACD~ CBD

A C B D

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click for Lab 1 - Similar Right Triangles Therefore, the altitude of a right triangle divides the triangle into two smaller triangles that are similar to the

• riginal triangle and similar to each other.

click

To prove this,

click

Teacher Notes

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A C B D

ABC~ ACD~ CBD

Let's prove the Theorem.

The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. Given: Prove: ABC is a right triangle is the altitude of ABC Statements Reasons

ABC is a right triangle is a right angle is a right angle is a right angle

ABC ~ ACD ABC ~ CBD ABC~ ACD~ CBD

Given Given

Def of Perp Lines. 2 lines that form a rt angle Def of Altitude All rt angles are Reflexive Prop of AA~ Def of Perp Lines All rt angles are Reflexive Prop of AA~ Transitive Prop of ~ click click click click click click click click click click click click

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Let's sketch the 3 triangle's separately, with the same orientation.

A C D

B C D

B A C Match up the angles.

Helpful tip: If you set , then you can assign all the angles a value and easily find the matches.

60 30

A C D

30 60 B C D 60 30

B A C

60 30 30 60

A C B D

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d e c b a

Because the triangles are similar the corresponding sides are proportional.

c B

A C

a b

A C D

d b

B C D

e a

ABC~ ACD~ CBD

Assign lengths to all the segments. Let the lengths of the segments on the hypotenuse be d and e. Label the sides of a triangle with the lower case letter of the opposite angle.

ABC~ ACD ABC~ CBD

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Statements Reasons

d e c b a

To prove the Pythagorean Theorem, use the proportions.

Given: Prove:

ABC is a right triangle.

is an altitude. ABC~ ACD ABC~ CBD

Using the multiplication property of equality, multiply the equation by bc. simplify Using the multiplication property of equality, multiply the equation by ac simplify Altitude of a rt triangle theorem. Altitude of a rt triangle theorem. Definition of similar triangles. Definition of similar triangles. (1) (2) click click click click click click click click

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Statements Reasons

d e c b a

To prove the Pythagorean Theorem, use the proportions (continued).

Given: Prove:

ABC is a right triangle.

is an altitude.

Distributive Property Simplify Using the addition property

• f equality, add equation (1)

and equation (2) together. Given Substitution click click click click click

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Example Find the length of the altitude KI?

H I J K 12 5 13

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It maybe helpful to sketch the 3 triangle's separately, with the same orientation.

J K I 5

H I K 12

H J K 5 12 13

H I J K 12 5 13

Because the triangles are similar the corresponding sides are proportional.

13x = 60 x ≈ 4.62

x x x

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5 P S Q 4 3 R

S Q 4

5 P R S x P Q R S 5 4 3 x

Try this... Find the length of RS. Answer

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15 Which ratio is the ratio of corresponding sides?

A B C D

H I J K

Answer

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16 Find KJ.

H I J K 7 24 25

Answer

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The next two theorems are Geometric Mean Theorems. What is a mean? An average. Usually when we ask to find the mean, we are asking for the arithmetic mean. What is an arithmetic mean? The sum of n values divided by the number of values (n). What is a geometric mean? The nth root of a product of n

• values. It is defined for only positive numbers (no

negative numbers, no zero) For more information click on this link: Arithmetic Mean vs Geometric Mean

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The geometric mean of two positive numbers a and b is the positive number x that satisfies a x x b x2 = ab x = = Visually, the geometric mean answers this question: given a rectangle with sides a and b, find the side of the square whose area equals that of the rectangle.

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Example Find the geometric mean of 8 and 14. x2 = 8(14) x2 = 112

(only the positive value)

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17 Find the geometric mean of 7 and 56. Write the answer is simplest radical form. A B C D

Answer

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18 Find the geometric mean of 3 and 48.

Students type their answers here

Answer

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Corollary The altitude drawn to the hypotenuse of a right triangle divides the the hypotenuse into two segments. The altitude is the geometric mean of the two segments formed. CD2 = AD(DB) CD is the altitude of ABC

Since, ACD~ CBD

A C B D

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8 z 6

Example Find z. Answer

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Example Find z.

18 z 6

Answer

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y 9 12

2)

18 8 y

1)

Try this... Find y. Answer

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19 Find x.

A B

100

C

20

D

50

5

x

10

Answer

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20 Find x.

A 99 B C D

11 9 x

Answer

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Corollary If the altitude drawn to the hypotenuse of a right triangle, divides the hypotenuse into two segments. The length of each leg of the original triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. AD AC AC AB = AB BC BC DB = CD is the altitude of ABC

A C B D

Since, ABC~ ACD~ CBD ABC~ ACD ABC~ CBD

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R

S T U 4

9 x

Example Find x. Answer

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D E F G 4 6 x

Example Find x. Answer

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21 Is PR a geometric mean between QR and SR?

True False

P Q R S

Answer

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22 Is the geometric mean correct?

True False

P Q R S

Answer

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23 Which proportion is correct?

A B C D

J

K L M

Answer

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24 Find y.

A B

20

C

5

D

12

9 16 y

Answer

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25 Find y.

A 3 B

18

C

24

D

None of the above

27 9

x y

Answer

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26 Find x.

5 8 x

Answer

Slide 73 / 240 Special Right Triangles

Return to the Table of Contents

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In this section you will learn about the properties of the two special right triangles.

45o 45o 90o 30o 60o 90o

45-45-90 30-60-90

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A 45-45-90 triangle is an isosceles right triangle, where the hypotenuse is √2 times the length of the leg. hypotenuse = leg(√2) 45o 45o

x√2

x x

45-45-90 Triangle Theorem

Can you prove this? Answer

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Example Find the length of the missing sides. Write the answer in simplest radical form. P Q R x y 45o 45o

6

Answer

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Example Find the length of the missing sides of the right triangle.

S T V 18 x y

Answer

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Try this... Find the length of the missing sides.

x y 8

Answer

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27 Find the value of x.

A 5 B

5√2

C

(5√2)/2

x y 5

Answer

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28 Find the value of y.

A 5 B

5√2

C

(5√2)/2

x y 5

Answer

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29 What is the length of the hypotenuse of an isosceles right triangle, if the length of the legs is 8 √2 inches.

Answer

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30 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm.

Answer

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In a 30-60-90 right triangle, the hypotenuse is twice the length of the shorter leg and the longer leg is √3 times the length of the shorter leg. hypotenuse = 2(shorter leg) longer leg = √3(shorter leg) x x√3 2x 30o 60o

30-60-90 Triangle Theorem Slide 84 / 240

This can be proved using an equilateral triangle. x x√3 2x 30o 60o

A C 60 60 30 30 c=2x 2x a=x x D b B

For right triangle ABD, BD is a perpendicular bisector. let a = x, c = 2x and b= BD

Slide 85 / 240

Example Find the length of the missing sides

• f the right triangle.

F G H 60o x y

30o

5

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F G H

60o x y

30o

5

Recall triangle inequality, the shortest side is

• pposite the smallest angle and the longest

side is opposite the largest angle. HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF Answer

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M A T 30o 60o x y 9 Example Find the length of the missing sides

• f the right triangle.

Answer

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Example Find the area of the triangle.

14 ft

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14 ft h

? ? The altitude (or height) divides the triangle into two 30o-60o-90o triangles. The length of the shorter leg is 7 ft. The length of the longer leg is 7√3 ft. A = b(h) = 14(7√3) A ≈ 84.87 square ft

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Try this... Find the length of the missing sides of the right triangle.

15

30o 60o

x y

Answer

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30o 9 ft

Try this... Find the area of the triangle. Answer

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30o 60o

x 7

31 Find the value of x.

A 7 B

7√3

C

(7√2)/2

D

14

Answer

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32 Find the value of x.

A 7 B

7√3

C

(7√2)/2

D

14

7√2 x

Answer

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33 Find the value of x.

A 7 B

7√3

C

(7√2)/2

D

14

30o 60o

x

7√3

Answer

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34 The hypotenuse of a 30

• -60o-90o triangle is 13 cm.

What is the length of the shorter leg?

Answer

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35 The length the longer leg of a 30

• -60o-90o triangle is

7 cm. What is the length of the hypotenuse?

Answer

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The wheelchair ramp at your school has a height of 2.5 feet and rises at angle of 30o. What is the length of the ramp?

Real World Example Slide 98 / 240

30o 2.5 ?

The triangle formed by the ramp is a 30

• -60o-90o right
• triangle. The length of the ramp is the hypotenuse.

hypotenuse = 2(shorter leg) hypotenuse = 2(2.5) hypotenuse = 5 The ramp is 5 feet long.

Slide 99 / 240

36 A skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45 . What is the height of the ramp? Round to the nearest hundredth.

45o 3 feet ?

Answer

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37 What is the length of the base of the ramp? Round to the nearest hundredth.

45o 3 feet ?

Answer

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38 The yield sign is shaped like an equilateral triangle. Find the length of the altitude.

20 inches

Answer

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39 The yield sign is shaped like an equilateral

• triangle. Find the area of the sign.

20 inches

Answer

Slide 103 / 240 Trigonometric Ratios

Return to the Table of Contents

Slide 104 / 240

Right triangle trigonometry is the study of the relationships between the sides and angles of right triangles.

a b c A B C

Slide 105 / 240

Leaning Tower of Pisa, Bell Tower in Pisa, Italy

Ever since the construction of the Bell Tower in the 1100's, it has slowly tilted south and is at risk of falling

• ver. If the angle of slant ever fall's

below 83 degrees, it is feared the tower will collapse.

Slide 106 / 240

A B C D F angle of slant GE

ABC~ DBE~ FBG WHY? Engineers can measure the angle of slant using any of the right triangles constructed below. Engineers very carefully measure the perpendicular distance from a tower window (points A, D or F) to the ground (points G, E or C). Then they measure the distance from the tower to points C, E or G. Answer

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Triangle Height Base Ratio Height / Base ABC AC=50m BC=5m 50/5=10 DBE DE=30m BE=3m 30/3=10 FBG FG=20m BG=2m 20/2=10

Let's calculate the ratio's of the height to the base for each right triangle. Notice that all of the ratios are the same. WHY? The ratio of height/base is also called the slope ratio (rise/run) or tangent ratio. Answer

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When the triangle is dilated (pull scale), how does the angle change? What happens to the slope ratio? What happens to the ratio when the angle increases? What happens to the ratio when the angle decreases? Click for interactive website to investigate.

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a b c A B C

To learn right triangle trigonometry, first you need to be able to identify the sides of a right triangle. In a right triangle, there are 2 acute

• angles. In the triangle to the left,

A and B are the acute angles.

Label the sides of a triangle with the lower case letter of the opposite angle.

Slide 110 / 240

Let's look at A, when A is the reference angle, the side opposite A is a. the side adjacent (or next to) A is b. and the hypotenuse is c.

a b c A B C

When B is the reference angle, the side opposite B is b. the side adjacent (or next to) B is a. and the hypotenuse is c.

a b c A B C

• pp

adj hyp

• pp

adj hyp

Slide 111 / 240

40 What is the side opposite to J?

A JL B

LK

C

KJ

J K L

Answer

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41 What is the hypotenuse of the triangle?

A JL B

LK

C

KJ

J K L

Answer

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42 What is the side adjacent to J?

A JL B

LK

C

KJ

J K L

Answer

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43 What is the side opposite K? A JL B LK C KJ

J K L

Answer

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44 What is the side adjacent to K? A JL B LK C KJ

J K L

Answer

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A trigonometric ratio is the ratio of the two sides of a right triangle. There are 3 ratios for each acute angle

• f a right triangle.

The ratios are called sine, cosine, and tangent (abbreviated sin, cos, and tan).

a b c A B C

Trigonometric Ratios Slide 117 / 240

sinθ =

• pposite side

hypotenuse cosθ =

adjacent side hypotenuse

tanθ =

• pposite side

adjacent side a b c A B C

The 3 Trigonometric Ratios

This spells....

SOHCAHTOA

• r

which is a pneumonic to help you remember the sides of a right triangle (you'll need to remember the spelling).

θ

Slide 118 / 240

Click for a SOHCAHTOA song on youtube.com "Gettin' Triggy Wit It".

Slide 119 / 240

Example Find the sin F, cos F, and tan F.

D E F 6 8 10

Since F is your reference angle, label the sides of the triangle opposite, adjacent and hypotenuse. Use the pneumonic to find the trig ratios. sinF = opp

hyp 6 10

=

3 5

= cosF = adj

hyp 8 10 = 4 5

= tan F = opp

adj 6 8

=

3 4

=

D E F 6 8 10

• pp

adj hyp

Always reduce fractions to lowest terms.

Slide 120 / 240

Example Find the sin D, cos D, and tan D.

D E F 6 8 10

Since D is your reference angle, label the sides of the triangle opposite, adjacent and hypotenuse. Use the pneumonic to find the trig ratios. sinD =opp

hyp 8 10 4 5

= = cosD = adj

hyp 6 10 = 3 5

= tanD = opp

adj 8 6

=

4 3

=

D E F 6 8 10

• pp

adj hyp

Always reduce fractions to lowest terms.

Slide 121 / 240

45 What is the sin R? A 20/29 B 21/20 C 21/29

D 20/21 Answer

Slide 122 / 240

46 What is the cosR?

A 20/29 B

21/20

C

21/29

D

20/21

Answer

Slide 123 / 240

47 What is the tanR?

A 20/21 B

21/20

C

20/29

D

21/29

Answer

Slide 124 / 240

48 What is the sinQ? A 20/29 B 21/20 C 21/29 D 29/20

Answer

Slide 125 / 240

49 What is the cosQ? A 20/29 B 21/20 C 21/29 D 29/21

Answer

Slide 126 / 240

50 What is the tanQ? A 20/29 B 21/20 C 21/29 D 20/21

Answer

Slide 127 / 240

The angle of slant of the Tower of Pisa is 84.3

A B C D F angle of slant

To find the trigonometric ratio of an angle, use a calculator or a trig table. Check that your calculator is set for degrees (not radians) and round your answer to the ten thousandth place (4 decimal places). Find the following: sin 84.3 = .9951 cos 84.3 = .0993 tan 84.3 = 10.0187

click click click

Slide 128 / 240

51 Evaluate sin 60. Round to the nearest ten thousandth. A 0.5 B 0.8660 C 1.7321 D 0.5774

Answer

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52 Evaluate cos 60. Round to the nearest ten thousandth. A 0.5 B 0.8660 C 1.7321 D 0.5774

Answer

Slide 130 / 240

53 Evaluate tan 60. Round to the nearest ten thousandth. A 0.5 B 0.8660 C 1.7321 D 0.5774

Answer

Slide 131 / 240

Trig tables were used by early mathematicians and astronomers to calculate distances that they were unable to measure directly. Today, calculators are usually used.

x

Slide 132 / 240

How do you find an unknown side measure in a right triangle when you are given an acute angle and one side?

• pp

adj You need to identify the correct trig function that will find the missing side. Use SOHCAHTOA to help. B is your angle of reference. Label the given and unknown sides of your triangle

• pp, adj, or hyp. Identify the trig funtion that uses

B, the unknown side and the given side. Using , I am looking for

• and I have a, so

the ratio is o/a which is tangent. now you can solve for x, the missing side.

Slide 133 / 240

B C

x

12 30o

A

Example Find the trig equation that will find x. Answer

Slide 134 / 240

B C

x

12 30o

A

Example Find the trig equation that will find x. Answer

Slide 135 / 240

B C

x

12 30o

A

Example Find the trig equation that will find x. Answer

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54 Using B, which is the correct trig equation needed to solve for x.

A sin40 = 12/x B

cos40 = x/12

C

tan40 = 12/x

D

sin40 = x/12

B E x 12 40o D

Answer

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55 Using D, which is the correct trig equation needed to solve for x.

A sin50 = 12/x B

cos50 = x/12

C

tan50 = 12/x

D

sin50 = x/12

B E x 12 50o D

Answer

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56 Using J, which is the correct trig equation needed to solve for x.

A tan32 = x/11 B

cos32 = x/11

C

tan32 = 11/x

D

sin32 = 11/x

J L K x 32o 11 Answer

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57 Using K, which is the correct trig equation needed to solve for x.

A tan58 = x/11 B

cos58 = x/11

C

tan58 = 11/x

D

sin 58 = 11/x

J L K x 58o 11 Answer

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Finding the Missing Side of a Right Triangle

• pp

adj Now, you can solve for x, the missing side. Round your answer to the nearest tenth.

Using your calculator, find the tan 84.3 Round your answer to 4 decimal places. You can rewrite 10.0187 with a denominator

• f 1 and use the cross product property or

multiply both sides of the equation by 5 using the multiplication property of equality (see next slide).

Slide 141 / 240

Finding the Missing Side of a Right Triangle

• pp

adj Now, you can solve for x, the missing side. Round your answer to the nearest tenth.

Multiply both sides of the equation by 5 using the multiplication property of equality.

Slide 142 / 240

12 G E M 25o x

Example Find x. Round your answer to the nearest hundredth. Answer

Slide 143 / 240

12 G E M 65o x Example Find x. Round your answer to the nearest hundredth. Answer

Slide 144 / 240

C A E y 10 20o

Example Find y. Round your answer to the nearest hundredth. Answer

Slide 145 / 240

58 Find the length of LM. Round your answer to the nearest tenth. L M P 12 68o

Answer

Slide 146 / 240

59 Find the length of LP. Round your answer to the nearest tenth. L M P 12 68o

Answer

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Explain and use the relationship between the sine and cosine of complementary angles.

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Find the measure of A?

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To find the measure of A... The sum of the interior angles of any triangle is equal to 180 degrees. A and B are complementary angles. Complementary angles are two angles whose sum of their measures is 90 degrees. The acute angles of a right triangle are always complementary.

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60 For right triangle ABC, what is the measure of B? A 30 degrees B 50 degrees C 60 degrees D cannot be determined A C B 30o

Answer

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61 If the , find the complementary angle? A 20 degrees B 70 degrees C 160 degrees D none of the above

Answer

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A C B

53.1 36.9 4 5 3

Let's compare the sine and cosine of the acute angles of a right triangle. In a right triangle, the acute angles are

• complementary. m A + m B = 90

53.1 + 36.9 = 90 sin A = 4/5 sin 53.1 = .7997 cos B = 4/5 cos 36.9 = .7997 sin A = cos B sin 53.1 = cos 36.9 The sine of an angle is equal to the cosine of its complement. cos A = 3/5 cos 53.1 = .6004 sin B = 3/5 sin 36.9 = .6004 cos A = sin B cos 53.1 = sin 36.9 The cosine of an angle is equal to the sine of its complement.

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L M P 12 68o 22o x

First, find the measure of LP using the sine function. Then, find the measure of LP using the cosine function. sine function cosine function Sine and Cosine are called co-functions of each other. Co-functions of complementary angles are equal.

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62 Given that sin 10 = .1736, write the cosine of a complementary angle. A sin 10 = .1736 B sin 80 = .9848 C cos 10 = .9848 D cos 80 = .1736

Answer

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63 Given that cos 50 = .6428, write the sine of a complementary angle. A sin 50 = .7660 B sin 40 = .6428 C cos 50 = .6428 D cos 40 = .7660

Answer

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64 Given that cos 65 = .4226, write the sine of a complementary angle. A sin 25 = .4226 B cos 25 = .9063 C sin 65 = .9063 D cos 65 = .4226

Answer

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65 What can you conclude about the sine and cosine of 45 degrees?

Students type their answers here

Answer

Slide 158 / 240 Solving Right Triangles

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Slide 159 / 240

To solve a right triangle means to find all 6 values in a triangle. The lengths of all 3 sides and the measures of all 3 angles.

a b c A B C

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Let's solve a right triangle given the length of one side and the measure of one acute an gle (AAS). You need to find the 3 missing parts.

C A B 15

x y z

64o

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C A B 15

x y z

64o

First, let's find the measure of A. Answer

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C A B 15

26 y z

64o

Then, let's find the measure of AB. Answer

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Then, let's find the measure of BC.

C A B 15

26 13.48

64o

z

Answer

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Try this... Find the missing parts of the triangle.

R E D 37o

11

Answer

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Let's solve a right triangle given the length of two sides (SSA).

9 15 A B C

x y z

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First, find the length of BC since we know how to do that. But, how do you find the measure of A and C?

9 15 A B C

x y z

Answer

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You will need to use the inverse trig functions. If sinθ = , θ = sin-1 If cosθ = , θ = cos-1 If tanθ = , θ = tan-1

a b c A B C

θ

With the sine, cosine and tangent trig functions, if you know the angle θ and the measure of one leg, then you can find the measure of a leg of a triangle. With the inverse sine, inverse cosine and inverse tangent trig functions, if you know the measures of 2 legs of a triangle, you can find the measure of the angle.

Pronounced inverse sine, inverse cosine, and inverse tangent.

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θ = sin-1( )

• pposite side

hypotenuse θ = cos-1( ) adjacent side hypotenuse θ = tan-1( )

• pposite side

adjacent side a b c A B C θ

The 3 Inverse Trigonometric Ratios

Use the inverse trig function to find the unknown angle measure when you know the length of 2 sides.

Remember:

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66 Find sin

• 10.8. Round the angle measure to the

nearest hundredth.

Answer

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67 Find tan

• 12.3. Round

the angle measure to the nearest hundredth.

Answer

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68 Find cos

• 10.45. Round

the angle measure to the nearest hundredth.

Answer

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9 15 A B C

θ

To find an unknown angle measure in a right triangle, You need to identify the correct trig function that will find the missing value. Use SOHCAHTOA to help. A is your angle of reference. Label the two given sides

• f your triangle opp, adj, or hyp.

Identify the trig funtion that uses A, and the two sides. Using , I have a and h, so the ratio is a/h which is cosine. now you can solve for A, the missing angle using the inverse trig function. adj hyp How are you going to calculate the measure of C? Answer

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B C

7

12

A

θ

Example Find the trig equation that will find θ. Answer

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B C 10

12

A

θ Example Find the trig equation that will find θ. Answer

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B C

9

12

A θ

Example Find the trig equation that will find θ. Answer

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69 Which is the correct trig equation to solve for A B C D

B E 7 12 D

Answer

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70 Which is the correct trig equation to solve for A B C D

B E 5 12 D

Answer

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71 Which is the correct trig equation to solve for A B C D

J L K 9 11 Answer

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Try this... Solve the right triangle. Round your answers to the nearest hundredth.

Q R

S 7

24

Answer

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72 Find CE.

C D E 8 5 Answer

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73 Find m C.

C D E 8 5 Answer

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74 Find the m E.

C D E 8 5 Answer

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75 Find the m G. L

A G

18 20o Answer

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76 Find AL. L A G

18 20o

Answer

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77 Find the m P.

A 49.19o B

33.69o

C

41.81o

D

56.31o

P E N 12 18 Answer

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78 Find RT.

A 10.44 B

12.45

C

11.47

D

9.53 40o S R T 8

Answer

Slide 187 / 240 Angle of Elevation and Depression

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Slide 188 / 240

How can you use trigonometric ratios to solve word problems involving angles of elevation and depression?

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When you look up at an

• bject, the angle your line
• f sight makes with a line

drawn horizontally is the angle of elevation.

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When you look down at an

• bject, the angle your line
• f sight makes with a line

drawn horizontally is the angle of depression.

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The angle of elevation and the angle of depression are both measured relative to parallel horizontal lines, they are equal in meaure.

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79 How can you describe the angle relationship between the angle of elevation and the angle of depression? A corresponding angles B alternate interior angles C alternate exterior angles D none of the above

Answer

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Example Amy is flying a kite at an angle of 58o. The kite's string is 1 58 feet long and Amy's arm is 3 feet off the ground. How high is the kite off the ground?

58o 1 5 8 f e e t

x 3 feet

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sinθ = sin58 = .8480 = x = 134

x 158 x 158 x 158

Now, we must add in Amy's arm height. 134 + 3 = 137 The kite is about 137 feet off the ground.

158ft x 58o

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30o 6 ft 5306 ft

x Example You are standing on a mountain that is 5306 feet high. You look down at your campsite at angle of 30

• . If you are 6 feet

tall, how far is the base of the mountain from the campsite?

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30o

x

5312 ft

tan30 = .5774 = .5774x = 5312 x ≈ 9,200 ft

5312 x 5312 x

The campsite is about 9,200 ft from the base of the mountain.

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Try this... You are looking at the top of a tree. The angle of elevation is 55o. The distance from the top of the tree to your position (line of sight) is 84 feet. If you are 5.5 feet tall, how far are you from the base of the tree?

Answer

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80 When you look down at an object, the angle your line of sight makes with a line drawn horizontally is the angle of _______.

A elevation B

depression

Answer

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81 Katherine looks down out of the crown of the statue

• f liberty to an incoming ferry about 345 feet. The

distance from crown to the ground is about 250 feet. What is the angle of depression?

Answer

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82 What is the distance from the ferry to the base of the statue?

Answer

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Law of Sines and Law of Cosines

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Slide 202 / 240

How can you solve a non-right triangle ? How can you find the side lengths and angle measures of non-right triangles? The Law of Sines and Law of Cosines can be used to solve any triangle. You can use the Law of Sines when you are given -

• 1. Two angle measures and any side length (AAS or ASA)
• 2. Two side lengths and the measure of a non-included angle

(SSA) when the angle is a right angle or an obtuse angle. The Law of Sines has a problem dealing with SSA when the angle is acute. There can be zero, one or two solutions. Click on: Khan Academy Video "More On Why SSA Is Not A Postulate" for more info. You can use the Law of Cosines when you are given -

• 3. Three side lengths (SSS)
• 4. Two side lengths and the measure of an

included angle (SAS)

Slide 203 / 240 Law of Sines

If ABC has sides of length a, b, and c, then sin A = sinB = sinC a b c To use the Law of Sines, 2 angles and 1 side must be given. a c A B C b

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Let's prove the Law of Sines

Given: Prove: ABC has sides of length a, b, and c

Statements Reasons

Given

Def of Altitude

If ABC has sides of length a, b, and c, then sin A = sinB = sinC a b c sin A = sinB = sinC a b c

a A C b c B

h Draw an altitude from C to side AB Let h be the length of the altitude ABC with side lengths a, b, and c

Def of sine Multiply by b. Mult Prop of =. Substitution Prop of = Divide by ab. Division Prop of = Multiply by a. Mult Prop of =. click click click click click click click

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Prove the Law of Sines (continued)

Given: Prove: ABC has sides of length a, b, and c

Statements Reasons

Def of Altitude

sin A = sinB = sinC a b c

Draw an altitude from B to side AC Let g be the length of the altitude

Def of sine Multiply by c. Mult Prop of =. Substitution Prop of = Divide by ac. Division Prop of = Multiply by a. Mult Prop of =.

a A b c B

g

C

Substitution Prop of = click click click click click click click

a A C b c B

h

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Use the Law of Sines to solve the triangle. A B

C 10 65o 70o a b

Select the ratios based on the given information. Given: m B, m C and BA (side c) (AAS) Which ratios must be used first? Why? sin A = sinB = sinC a b c Answer

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A B C 10 65o 70o a b

sinB = sinC b c sin70 = sin65 b 10 First we can find the length side b. Answer

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Before we find the length of side a , we find the m A .

A B 10 65o 70o a b=10.37 C

Triangle Sum Theorem m A + m B + m C = 180o Answer

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A C 10 65o 70o a b=10.37 =45o B sinA = sinC a c Now we find the length side a. Answer

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Try this... Use the Law of Sines to find the length of side b (ASA).

a b 9 85o 29o A B C Since the length of the side opposite <C is given, find the m<C first.

hint

Answer

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Example... Find the length of side b (SSA with an obtuse angle).

2.8

101o

A B 8 b C Answer

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83 Find the m A.

A 19o B

31o

C

29o

D

28o

A B C 10 70o 81o b c

Answer

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84 Which ratio must be used to find the length of b or c?

A sinA

a

B

sinA b

C

sinB b

D

sinC c

A B C 10 70o 81o b c

Answer

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85 What is the length of b?

A B C 10 70o 81o b c

Answer

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86 What is the length of c?

A B C 10 70o 81o b c

Answer

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a c A B C b

If ABC has sides of length a, b, and c, then: To use the Law Cosines, you must be given the length of 3 sides (SSS) or the length of 2 sides and the measure of the included angle (SAS).

Law of Cosines Slide 217 / 240

Let's prove the Law of Cosines

Given: Prove: ABC has sides of length a, b, and c

Statements Reasons

Given Def of Altitude

If ABC has sides of length a, b, and c, then

a A C b

h

D x c-x B c

Draw an altitude CD from C to side

• AB. Let h be the length of the alt.

ABC with side lengths a, b, and c

Segment Addition Postulate Multiply by b. Mult Prop of =. Simplify Substitution, equation (2)

a A C b c B

Let x be the length of AD. Then (c-x) is the length of DB. In ADC, cosA = x/b x=b(cosA)

Definition of cosine

In ADC,

Pythagorean Theorem

In CDB,

Pythagorean Theorem

(1) (2)

Associative Prop of Addition Substitution, equation (1)

(similar reasoning shows that ) click click click click click click click click click click click

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a is opposite <A b is opposite <B c is opposite <C

Example Use the Law of Cosines to solve the right triangle.

A B C c=27 a=16 b=23

The formula you choose depends on which angle you are solving for first.

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To find the m A, a2 = b2 + c2 - 2bc(cosA) 162 = 232 + 272 - 2(23)(27)(cosA) 256 = 529 + 729 - 1242(cosA) 256 = 1258 - 1242(cosA)

• 1002 = -1242(cosA)

.8068 = cosA A = cos-1(.8068) m A ≈ 36.22o

A B C c=27 a=16 b=23

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To find the m B, b2 = a2 + c2 -2ac(cosB) 232 = 162 + 272 - 2(16)(27)(cosB) 529 = 256 + 729 - 864(cosB) 579 = 985 - 864(cosB)

• 406 = -864(cosB)

.4699 = cosB B=cos-1(.4699) m B ≈ 61.97o

A B C c=27 a=16 b=23 36.22

• r

Using 2 different methods, the answers are slightly different because of rounding.

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To find the m C, Use the Triangle Sum Theorem.

A B C c=27 a=16 b=23 36.22

61.97

Answer

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Try this... Use the Law of Cosines to find the m<B (SSS).

A B C 5 6 7

Answer

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87 In the triangle the length of c is...

A 8 B

9

C

15

A B C 8 9 15

Answer

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88 In the triangle the length of a is...

A 8 B

9

C

15

A B C 8 9 15

Answer

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89 Which formula would you use to find the m<A?

A a2 = b2 + c2 - 2ac(cosA) B

a2 = b2 + c2 - 2bc(cosA)

C

b2 = a2 + c2 - 2ac(cosB)

D

c2 = a2 + b2 - 2ab(cosC)

Answer

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90 What is the m A?

A B C 8 9 15

Answer

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91 What is the m C?

A B C 8 9 15

Answer

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92 What is the measure of B (ASA)?

Students type their answers here

Answer

B A C 4 8 50

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93 The Law of Sines and Cosines is used to solve...

A right triangles B

acute triangles

C

• btuse triangles

D

all triangles

Answer

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Return to the Table of Contents

Area of an Oblique Triangle Slide 231 / 240

Do you remember this? Previously, we found the area of a triangle when we were given 3 sides. Find the area of the triangle.

13 feet 10 feet 13 feet

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b is the base of the triangle b = 10. h is the altitude (or height). It is the perpendicular bisector of the base in an isosceles triangle.

13 feet 5 feet 13 feet

h

5 feet

Find h, using the pythagorean theorem - A = bh

1 2

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What formula can you use to find the area of a triangle if you know the length of two sides and the measure of an included angle (SAS)? Find the area of the triangle.

10 feet 13 feet

67.38

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10 feet 13 feet

h

67.38

Since A = bh and b = 10, we need to find h.

1 2

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Let's derive the formula for an oblique triangle.

Given: Prove:

Statements Reasons

Given

Def of Altitude

ABC has sides of length a, b, and c. Altitude h.

a A C b c B

h Draw an altitude from B to side AC Let h be the length of the altitude ABC with side lengths a, b, and c

Def of sine Substitution Prop of =

• Definition. Formula for

the area of a triangle. Multiply by a. Mult Prop of =. Commutative Prop of Multiplication click click click click click click click

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94 Which of the following expressions can be used to find the area of the triangle below? Select all that apply. A B C D E F Answer

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95 Find the area of the triangle to the nearest tenth.

Students type their answers here

Answer

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96 Find the area of the triangle to the nearest tenth.

Students type their answers here

Answer

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97 Find the area of the triangle to the nearest tenth.

Students type their answers here

Answer

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