SECTION Ready To Go On? Skills Intervention 8A 8-1 Similarity in Right Triangles · 2014-09-16 · 8-1 Similarity in Right Triangles ... m C 90 The acute angles of a right triangle

Copyright by Holt, Rinehart and Winston. 109 Holt GeometryAll rights reserved.

Find this vocabulary word in Lesson 8-1 and the Multilingual Glossary.

Finding Geometric Means

The geometric mean of two positive numbers is the positive square root of

their .

Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.

A. 4 and 16

Let x be the geometric mean.

x 2 (4)( ) Definition of geometric mean.

x 2

x Find the positive square root.

B. 8 and 25

Let x be the geometric mean.

x 2 (8)( ) Definition of geometric mean.

x

200

Find the positive square root.

Finding Side Lengths in Right TrianglesFind x, y, and z. z x

y

168

x 2 (8)(16) x is the geometric mean of 8 and .

x

128

2 Find the positive square root.

y 2 (24)( ) 384 y is the geometric mean of 24 and .

y

384

6 Find the positive square root.

z 2 ( )(8) z is the geometric mean of and 8.

z

8

Find the positive square root.

Name Date Class

Ready To Go On? Skills Intervention8-1 Similarity in Right Triangles8A

SECTION

Vocabulary

geometric mean

The geometric mean of two positive numbers is the positive square root of their product.

To estimate the height of a lighthouse, Henry stands so that his lines of sight to the top and bottom of the lighthouse form a 90 angle. What is the height of the lighthouse x to the nearest foot?

5 ft

24 ft 9 in.

x

Understand the Problem

1. How tall is Henry?

2. What forms the 90 angle? and

Make a Plan

3. What do you need to determine?

4. Since the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse, what is

the geometric mean of 5 and x ?

5. How many feet is 9 inches?

6. 24 ft 9 inches ft

7. Let x represent the height of the lighthouse above eye level.

(length of the altitude ) 2 geometric mean of 5 and x.

(24.75 ) 2 x

Solve

8. Solve the equation by isolating x. 9. The height of the tower to the nearest

(24.75 ) 2 5x foot is ft.

612.5625 ________ x

ft x

Look Back

10. Substitute your solution for x into the equation you wrote in Exercise 7.

(24.75 ) 2 5(123)

612.5625

11. Is your solution approximately equal to square of the length of the altitude?

8AReady To Go On? Problem Solving Intervention8-1 Similarity in Right Triangles


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SECTION

Find these vocabulary words in Lesson 8-2 and the Multilingual Glossary.

Using Trigonometric Ratios to Find Lengths

sin A opposite leg __________ hypotenuse cos A _____________ hypotenuse tan A

_____________ adjacent leg

Find each length. Round to the nearest hundredth.

A. LM

_

LM is to the given angle, K.

You are given KM, which is to K. 4035 in. K

L

M Since the adjacent and opposite legs are involved, use the

ratio to find LM.

tan K opposite leg __________ adjacent leg LM ______ Write the trigonometric ratio.

tan LM ______ Substitute the given values.

( ) tan 40 Multiply each side by .

in. LM Simplify the expression.

B. XZ

_

XZ is to the given angle, Y.

You are given YZ, which is the of the triangle.

Since the opposite side and the hypotenuse are involved, 2615.3 cm

X

Y

Z

use the ratio to find XZ.

sin Y opposite leg __________ hypotenuse ______ 15.3 Write the trigonometric ratio.

sin XZ ______ Substitute the given values.

( ) sin 26 Multiply each side by .

cm XZ Simplify the expression.

8AReady To Go On? Skills Intervention8-2 Trigonometric Ratios

Name Date Class


SECTION

Vocabulary

trigonometric ratio sine cosine tangent


Solving Right Triangles

Given measures can be used to find unknown angle measures or lengths of a triangle, which is known as solving a triangle. To solve a right triangle,

you need to know two side lengths or one side length and a(n) angle measure.

Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

AB

C

6.8

3.2Method 1

By the Pythagorean Theorem,

A C 2 A B 2 B C 2

(6.8 ) 2 ( ) 2 Substitute the given values.

AC

AC Find the square root.

mA ta n 1 (3.2) ______ If tan A x, then ta n 1 x mA.

mA

mC 90 The acute angles of a right triangle are complementary.

Method 2

mA ta n 1 _______ 6.8 If tan A x, then ta n 1 x mA.mC 90 The acute angles of a right triangle are

.

sin A 3.2 ___ AC Definition of Sine.

AC _______ sin A Solve for AC.

AC 3.2 ______________________

sin

Substitute for mA.

AC

Ready To Go On? Skills Intervention8-3 Solving Right Triangles

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8A


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You can use the inverse tangent ratio to find the measure of unknown angle measures. If tan A x, then ta n 1 x mA.

A carpenter frames a garage which has a roof that peaks at 8 feet from the ceiling joist. If the length of the ceiling joist is 12 feet from the center point to the edge of the garage, what angle is formed by the rafter and the ceiling joist? Round to the nearest degree.


1. What is the height from the ceiling joist to the peak of the roof?

2. Are the ceiling joists horizontal or vertical?

3. How long is the ceiling joist from the edge to the center of the garage.

Make a Plan

4. What do you need to determine?

5. Label the right triangle to show the garage roof.

A

B

Cpeak

rafter

height = ft

ceiling joist length = ft

6. What angle is formed by the rafter and the ceiling joist?

7. Which trigonometric function involves opposite and adjacent legs of a triangle?

Solve mA ta n 1 _____ 12 8. If tan A x, then ta n 1 x mA. Complete: mA

9. What angle is formed by the rafter and the ceiling joist?

Look Back

10. Is tan 34 approximately equal to 8 ___ 12 ? 11. Does your answer seem reasonable? Explain.

Ready To Go On? Problem Solving Intervention8-3 Solving Right Triangles8A

SECTION

Copyright by Holt, Rinehart and Winston. 114 Holt GeometryAll rights reserved.All rights reserved.

8-1 Similarity in Right TrianglesFind the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form.

1. 6 and 3 2. 5.5 and 88 3. 5 __ 4 and 15 ___ 8

Find x, y, and z.

4. 5. 6.

z

xy

7

18

z

x y 18

12

z

x

y

24

12

x x x

y y y

z z z

7. A mountain climbing instructor is setting up a practice mountain and needs to know how much rope he will need. He positions himself so that his line of sight to the top of the cliff and his line of sight to the bottom form a right angle as shown. What is the height h of the practice mountain?

18 ft

5 ft

h

12

8-2 Trigonometric RatiosUse a special right triangle to write each trigonometric ratio as a fraction.

8. sin 45 9. cos 45 10. tan 30

8AReady To Go On? Quiz

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Copyright by Holt, Rinehart and Winston. 115 Holt Geometry

Ready To Go On? Quiz continued

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

11. sin 32 12. cos 84 13. tan 54

Find each length. Round to the nearest hundredth.

14. CB 15. PQ 16. JK

A

BC37

6

65

8

P Q

R

J K

L

226

8-3 Solving Right TrianglesFind the unknown measures. Round lengths to the nearest tenth and angle measures to the nearest degree.

17. 18. 19.

A

BC34

7.8

20

12

L

M

N

25

13

Z

X Y

A N Z

CB LN XZ

AC MN XY

20. The wheel chair ramp at a local arena has a ramp length of 20 feet and a rise of 1.7 inches. What angle does the ramp make with the sidewalk? Round to the nearest tenth of degree.

8A

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Ready To Go On? Enrichment

Trigonometry and Bearings

In surveying and navigation, directions are usually given in terms of bearings. A bearing measures the acute angle a path or line of sight makes with a fixed north-south line. The following examples explain this concept.

37

E

N

S

W

S 37 E means37 degreeseast of south

75E

N

S

W

N 75 W means75 degreeswest of north

60E

N

S

W

N 60 E means60 degreeseast of north

Solve

1. Two lighthouses are 30 miles apart, 2. A hot air balloon is tethered due west lighthouse A being due west of of an observation station. The landing pad lighthouse B. A boat is spotted from is 12 km south of the hot air balloon. the towers, and the bearings from A and From the landing pad, the bearing to the B are E 14 N and W 34 N, respectively. observation station is N 6320 E. How far Find the distance d of the boat from the is the hot air balloon from the observation line segment AB. station?

A Bd

14 34

30 miles

6320'12 km

pad

3. A ship leaves port at 8 A.M. and has a 4. A radio-controlled airplane is 160 meters bearing of S 29 W. If the ship sails at north and 85 meters east of the pilot. If 20 knots, how many nautical miles south the pilot wants the plane to fly directly to and how many nautical miles west will him, what bearings should be taken? the ship have travel by 2 P.M.?

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8A



Classifying Angles of Elevation and DepressionClassify each angle as an angle of elevation or angle of depression.

An angle of elevation is the angle formed by a horizontal line and a line of sight to a point

the line.

An angle of depression is the angle formed by a horizontal line and a line of sight to a point

the line.

DB

A

C

A A is formed by a horizontal line and a line of sight to a point

the line. It is an angle of .

B B is formed by a horizontal line and a line of sight to a point


C C is formed by a horizontal line and a line of sight to a point


D D is formed by a horizontal line and a line of sight to a point


Finding a Measure Using Trigonometric RatiosFind the missing measure x.

What are the ratios for each trigonometric ratio?

A

BC42

150

xsin A opposite ____________ cos A ____________ hypotenuse tan A

opposite ____________

What is _

AC in relation to B ?

What is _

CB in relation to B ?

Which trigonometric ratio can be used to find x ?

Set up the ratio: tan ______ x

x 150 _________ tan

The length of _

CB is approximately units long.

8BReady To Go On? Skills Intervention8-4 Angles of Elevation and Depression

Name Date Class

SECTION

Vocabulary

angle of elevation angle of depression


Determine if you have an angle of elevation of depression before starting to solve a problem.

A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 72. How tall is the tree? Round to the nearest foot.

7250 ft


1. What are you asked to find?

2. Do you have an angle of elevation or depression?

3. The triangle formed is a right triangle because the forms a

90 angle with the .

Make a Plan

4. You need to determine the side the given angle.

5. In relation to the given angle which side measure are you given?

6. Which trigonometric ratio can be used to find the height of the tree?

Solve

7. Set up the trigonometric ratio: 72 x ______

8. Solve the ratio for x. 72 x ______

x tan 72

x

9. The height of the tree is approximately ft.

Look Back

10. Since you have the measures of two sides, you can use the two sides and check to see if the measure of the given angle matches your calculation.

tan ______ 50

tan

tan 1

Does your result match the given angle measure, 72?

8BReady To Go On? Problem Solving Intervention8-4 Angles of Elevation and Depression

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Using the Law of SinesThe Law of Sines can be used to solve a triangle if you are given: two angle measures and any side length (ASA or AAS) or two side lengths and a non-included angle measure (SSA).

Find BC. Round to the nearest tenth.

Looking at the given diagram, are you given ASA, AAS or SSA?

State the Law of Sines: sin A ____ a ______ b ______

Which two proportions should be used?

B

A

C

11542

36 sin A ____ a sin B _______

sin __________ BC sin

__________ 36 Substitute known value.

BC sin sin 42 Use the Cross Products Property.

BC sin 42 ____________ sin

Divide to solve for BC.

BC The length of side BC is .

Using the Law of Cosines

The Law of Cosines can be used to solve a triangle if you are given: two side lengths and the included angle measure (SAS) or three side lengths (SSS).

B

A

C

62

17

12

Find BC. Round to the nearest tenth.

Looking at the given diagram, are you given SAS, or SSS?

Which formula for the Law of Cosines should be used?

a 2 c 2 2 c cos

What does b equal? What does c equal?

Substitute known values into the formula.

a 2 1 2 2 2( )( ) cos 62

a 2

a The length of _

BC is .

8BReady To Go On? Skills Intervention8-5 Law of Sines and Law of Cosines

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Finding the Magnitude of a VectorDraw vector 3, 3 on a coordinate plane. Find its magnitude to the nearest tenth.

Use the origin as the initial point. Then (3, 3) is the

point.

What is the horizontal change?

What is the vertical change?

The of a vector is its length.

To find the magnitude you use the Distance Formula.

24

y

x

4

2

2

4

2 4

3, 3

( 0 ) 2 ( 0 ) 2

9

Finding the Direction of a VectorThe force exerted by a skier on a tow rope is given by the vector 4, 5.Draw the vector on a coordinate plane.Find the direction of the vector to the nearest degree.

Step 1 Draw the vector on the coordinate plane.

Use the origin as the initial point. Then (4, 5) is the

point. 24

y

x

4

2

2

4

2 4

What is the horizontal change?

What is the vertical change?

Step 2 Find the direction. Label the drawing.

The angle you are looking for is formed by the vector and the x-axis.

Which trigonometric formula will be used?

Complete: tan A ______ 4

mA ta n 1 ______ 4

Ready To Go On? Skills Intervention8-6 Vectors

Name Date Class

SECTION

8B

Vocabulary

vector component form magnitude direction

equal vectors parallel vectors resultant vector


Name Date Class

The component form of a vector lists the horizontal and vertical change from the initial point to the terminal point.

To reach a campsite, a hiker first walks for 3 miles at a bearing of N 50 E. She then walks 5 miles due east. How far is the hiker from where she started and what direction? Round speed to the nearest tenth and the direction to the nearest degree.


1. What are you asked to find?

Make a Plan

2. Write the vector in form for the hiker and the

resultant .

Solve

3. Complete the vector sketches.

503

y

x 5W

N

S

E

N

W E

S

First Walk Second Walk

4. What angle does the vector in the First Walk make with the x-axis? 90 50

5. Write the vector for the hiker in component form. cos 40 x __ 3 , so x 3 cos 40 .

sin 40 y __ 3 , so y sin 40 . The hikers vector is , 1.9.

6. Write the vector for traveling east in component form. , 0

7. Find the resultant vector: , 1.9 , 0 , 1.9

8. Find the magnitude of the resultant vector.

(7.3 0 ) 2 ( 0 ) 2

9. The angle measure formed by the resultant vector gives the actual direction.

tan A ______ 7.3 , so A ta n 1 _______ 7.3 or N 75 E.

Look Back

10. Plot your findings on the graph. Does your result make sense?

Ready To Go On? Problem Solving Intervention8-6 Vectors8B

SECTION


8-4 Angles of Elevation and Depression

1. The scout at the top of a 1800-ft mountain spots a campsite. He measures the angle of depression to be 33. How far is the campsite from the foot of the mountain? Round to the nearest foot.

33

1800 m

2. The angle of elevation from a ship to the top of a lighthouse is 4. If the ship is 1200 m from the lighthouse, how tall is the lighthouse?

4

1200 m

8-5 Laws of Sines and Laws of CosinesFind each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.

3. AB 4. EG 5. AB

B

A

C

40

24

67

E

F

G

43

57

5

B

A C23

10131

6. AB 7. G 8. CB

B

A C48

32

126

E

F

G8

9

7

B

A

C

12

2430

8BReady To Go On? Quiz

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Ready To Go On? Quiz continued

8-6 VectorsDraw each vector on a coordinate plane. Find its magnitude to the nearest tenth.

9. 6, 2 10. 5, 2 11. 1, 4

y

x

4

2

2

4

62 4

24

y

x

2

2

4

6

2 4

24

y

x

2

2

4

6

2 4

Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree.

12. A wind velocity is given by the vector 13. The path of a hiker is given by the 3, 2. vector 6, 4.

24

y

x

2

2

4

6

2 4

y

x

2

2

4

6

2 4 6

14. The velocity of a plane is given by the vector 5, 5.

y

x

2

2

4

6

2 4 6

15. A canoeist leaves shore at a bearing of N 50 E and paddles at a constant speed of 6 mi/h. There is a 2 mi/h current moving due east. What are the canoes actual speed and direction? Round the distance to the nearest tenth of a mile and the direction to the nearest degree.

8B

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SECTION

Law of Sines and Law of Cosines

Solve each problem. Sketch a diagram and identify whether you will use the Law of Sines or Law of Cosines.

1. To find the distance between two points A and B on opposite sides of a small pond, a surveyor determines that AC is 107.5 feet, angle ACB is 56.2, and angle CAB is 78.6. Find the distance between AB. Round to the nearest tenth.

2. To find the distance XY across a canyon the following measurements were taken. XZ is 570 yards, angle YXZ is 103.4 and angle XZY is 34.6.

X

Y

Z

3. A surveyor is trying to determine the distance between points A and B. His view is obstructed by a large barn. He determines that CA is 75 feet, CB is 58 feet and angle ACB is 83. Find AB to the nearest foot.

4. A grove of trees is obstructing the view of a surveying crew. They have determined that the following distances: AC 143 feet, BC 125 feet and the measure of angle ACB is 82.4. What is the measure of BA? Round to the nearest foot.

5. A boat race starts at point J proceeds to point K and then point L before returning to the starting point. Using the diagram shown, determine the total distance of the race.

53

42

9 km

J

L

K


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SECTION Ready To Go On? Skills Intervention 8A 8-1 Similarity in Right Triangles · 2014-09-16 · 8-1 Similarity in Right Triangles ... m C 90 The acute angles of a right triangle