Simplifying Radical Expressions - Woodland Hills School ... 12 Ch 11 practic… · Simplifying Radical Expressions ... products you would use to simplify the expression (2 ... 11.
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Skills Practice
Simplifying Radical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
Pre-Activity How are radical expressions used in space exploration?
Read the introduction to Lesson 11-1 at the top of page 586 in your textbook.
Suppose you want to calculate the escape velocity for a spacecraft taking offfrom the planet Mars. When you substitute numbers in the formula, whichnumber is sure to be the same as in the calculation for the escape velocityfor a spacecraft taking off from Earth?
Reading the Lesson
1. a. How can you tell that the radical expression Ï28x2yw4w is not in simplest form?
b. To simplify Ï28x2yw4w, you first find the of 28x2y4.
You then apply the . In this case,
Ï4 ? 7 ?w x2? yw4w is equal to the product . You can simplify
again to get a final answer of 2|x|y2Ï7w.
2. Why is it correct to write Ïy4w 5 y2, with no absolute value sign, but not correct to write
Ïx2w 5 x?
3. What method would you use to simplify ?
4. What should you do to write the conjugate of a binomial of the form aÏbw 1 cÏdw? To
write the conjugate of a binomial of the form aÏbw 2 cÏdw?
Helping You Remember
5. What should you remember to check for when you want to determine if a radicalexpression is in simplest form?
Ï12tw}
Ï15w
Skills Practice
Operations with Radical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
Pre-Activity How can you use radical expressions to determine how far a personcan see?
Read the introduction to Lesson 11-2 at the top of page 593 in your textbook.
Suppose you substitute the heights of the Sears Tower and the Empire StateBuilding into the formula to find how far you can see from atop each building.What operation should you then use to determine how much farther you cansee from the Sears Tower than from the Empire State Building?
Reading the Lesson
1. Indicate whether the following expressions are in simplest form. Explain your answer.
a. 6Ï3w 2 Ï12w
b. 12Ï6w 1 7Ï10w
2. Below the words First terms, Outer terms, Inner terms, and Last terms, write the
products you would use to simplify the expression (2Ï15w 1 3Ï15w)(6Ï3w 2 5Ï2w).
First terms Outer terms Inner terms Last terms
1 1 1
Helping You Remember
3. How can you use what you know about adding and subtracting monomials to help youremember how to add and subtract radical expressions?
Skills Practice
Radical Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
Find the length of each missing side. If necessary, round to the nearest
hundredth.
1. 2. 3.
If c is the measure of the hypotenuse of a right triangle, find each missing
measure. If necessary, round to the nearest hundredth.
4. a 5 24, b 5 45, c 5 ? 5. a 5 28, b 5 96, c 5 ?
6. b 5 48, c 5 52, a 5 ? 7. c 5 27, a 5 18, b 5 ?
8. b 5 14, c 5 21, a 5 ? 9. a 5 Ï20w, b 5 10, c 5 ?
10. a 5 Ï75w, b 5 Ï6w, c 5 ? 11. b 5 9x, c 5 15x, a 5 ?
Determine whether the following side measures form right triangles. Justify your
answer.
12. 11, 18, 21 13. 21, 72, 75
14. 7, 8, 11 15. 9, 10, Ï161w
16. 9, 2Ï10w, 11 17. Ï7w, 2Ï2w, Ï15w
18. STORAGE The shed in Stephan’s back yard has a door that measures 6 feet high and 3 feet wide. Stephan would like to store a square theater prop that is 7 feet on a side.Will it fit through the door diagonally? Explain.
SCREEN SIZES For Exercises 19–21, use the following information.
The size of a television is measured by the length of the screen’s diagonal.
19. If a television screen measures 24 inches high and 18 inches wide, what size television is it?
20. Darla told Tri that she has a 35-inch television. The height of the screen is 21 inches.What is its width?
21. Tri told Darla that he has a 5-inch handheld television and that the screen measures 2 inches by 3 inches. Is this a reasonable measure for the screen size? Explain.
124
b1911
a
60
32c
Practice
The Pythagorean Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
11-411-4
Reading to Learn Mathematics
The Pythagorean Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
Pre-Activity How is the Pythagorean Theorem used in roller coaster design?
Read the introduction to Lesson 11-4 at the top of page 605 in your textbook.
The diagram in the introduction shows a right triangle and part of theroller coaster. Which side of the right triangle has a length approximatelyequal to the length of the first hill of the roller coaster?
Reading the Lesson
Complete each sentence.
1. The words leg and hypotenuse refer to the sides of a triangle.
2. In a right triangle, each of the two sides that form the right angle is a of the right triangle.
3. The longest side of a right triangle is called the of the right triangle.
Write an equation that you could solve to find the missing side length of each
right triangle.
4. 5. 6.
7. Suppose you are given three positive numbers. Explain how you can decide whetherthese numbers are the lengths of the sides of a right triangle.
Helping You Remember
8. Think of a word or phrase that you can associate with the Pythagorean Theorem to helpyou remember the equation c2
5 a21 b2.
9
106
109
10
Skills Practice
The Distance Formula
NAME ______________________________________________ DATE ____________ PERIOD _____
Find the distance between each pair of points whose coordinates are given.
Express answers in simplest radical form and as decimal approximations rounded
to the nearest hundredth if necessary.
1. (4, 7), (1, 3) 2. (0, 9), (27, 22)
3. (4, 26), (3, 29) 4. (23, 28), (27, 2)
5. (0, 24), (3, 2) 6. (213, 29), (21, 25)
7. (6, 2), 14, 2 8. (21, 7), 1 , 62
9. 12, 2 2, 11, 2 10. 1 , 212, 12, 2
11. (Ï3w, 3), (2Ï3w, 5) 12. (2Ï2w, 21), (3Ï2w, 3)
Find the possible values of a if the points with the given coordinates are the
indicated distance apart.
13. (4, 21), (a, 5); d 5 10 14. (2, 25), (a, 7); d 5 15
15. (6, 27), (a, 24); d 5 Ï18w 16. (24, 1), (a, 8); d 5 Ï50w
17. (8, 25), (a, 4); d 5 Ï85w 18. (29, 7), (a, 5); d 5 Ï29w
BASEBALL For Exercises 19–21, use the following information.
Three players are warming up for a baseball game. Player B stands 9 feet to the right and 18 feet in front of Player A.Player C stands 8 feet to the left and 13 feet in front of Player A.
19. Draw a model of the situation on the coordinate grid.Assume that Player A is located at (0, 0).
20. To the nearest tenth, what is the distance between Players Aand B and between Players A and C?
21. What is the distance between Players B and C?
22. MAPS Maria and Jackson live in adjacent neighborhoods. If they superimpose acoordinate grid on the map of their neighborhoods, Maria lives at (29, 1) and Jacksonlives at (5, 24). If each unit on the grid is equal to approximately 0.132 mile, how farapart do Maria and Jackson live?
A(0, 0)
B(9, 18)
C(28, 13)
x
y
O 4 8
16
12
8
4
28 24
1}3
2}3
1}2
1}2
1}3
1}2
Practice
The Distance Formula
NAME ______________________________________________ DATE ____________ PERIOD _____
11-511-5
Reading to Learn Mathematics
The Distance Formula
NAME ______________________________________________ DATE ____________ PERIOD _____
Pre-Activity How can the distance between two points be determined?
Read the introduction to Lesson 11-5 at the top of page 611 in your textbook.
What are the coordinates of points A, B, and C?
Reading the Lesson
1. Suppose you want to use the Distance Formula to find the distance between (6, 4) and (2, 1). Use (x1, y1) 5 (6, 4) and (x2, y2) 5 (2, 1). Complete the equations by writing the correct numbers in the blanks.
a. x1 5 y1 5 x2 5 y2 5
b. d 5 Ïwwww( 2 )21 ( 2 )2
2. Suppose you want to use the Distance Formula to find the distance between (3, 7) and(9, 22). Use (x1, y1) 5 (3, 7) and (x2, y2) 5 (9, 22). Complete the equations by writing thecorrect numbers in the blanks.
a. x1 5 y1 5 x2 5 y2 5
b. d 5 Ïwwww( 2 )21 ( 2 )2
3. A classmate is using the Distance Formula to find the distance between two points. She
has done everything correctly so far, and her equation is d 5 Ï(22 2w 5)21w (7 2w11)2w.
This equation will give her the distance between what two points?
Helping You Remember
4. Sometimes it is easier to remember a formula if you can state it in words. How can youstate the Distance Formula in easy-to-remember words?
x
y
O
A
B
Skills Practice
Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
Determine whether each pair of triangles is similar. Justify your answer.
1. 2.
For each set of measures given, find the measures
of the missing sides if nABC , nDEF.
3. c 5 4, d 5 12, e 5 16, f 5 8
4. e 5 20, a 5 24, b 5 30, c 5 15
5. a 5 10, b 5 12, c 5 6, d 5 4
6. a 5 4, d 5 6, e 5 4, f 5 3
7. b 5 15, d 5 16, e 5 20, f 5 10
8. a 5 16, b 5 22, c 5 12, f 5 8
9. a 5 , b 5 3, f 5 , e 5 7
10. c 5 4, d 5 6, e 5 5.625, f 5 12
11. SHADOWS Suppose you are standing near a building and you want to know its height.The building casts a 66-foot shadow. You cast a 3-foot shadow. If you are 5 feet 6 inchestall, how tall is the building?
12. MODELS Truss bridges use triangles in their support beams. Molly made a model of atruss bridge in the scale of 1 inch 5 8 feet. If the height of the triangles on the model is4.5 inches, what is the height of the triangles on the actual bridge?
11}2
5}2
D F
E
e
f d
A C
B
b
c a
808
478478
568
E H
F
GD
C
318 598R Q S T
UP
Practice
Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
11-611-6
Reading to Learn Mathematics
Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
11-6Pre-Activity How are similar triangles related to photography?
Read the introduction to Lesson 11-6 at the top of page 616 in your textbook.
How would you describe the shapes and sizes of the figures in the diagram?
Reading the Lesson
Complete each sentence.
1. In similar triangles, the angles of the two triangles can be matched so that
angles have equal .
2. If the angles of one triangle do not have the same measures as the angles of a second
triangle, then the two angles are not .
3. If two triangles have the same size and shape, then the measures of the corresponding
sides are .
Determine whether each pair of triangles is similar. Explain how you know that
your answer is correct.
4.
5.
6.
Helping You Remember
7. How can you use the idea that the corresponding sides of similar triangles areproportional to help you remember how to find the unknown lengths of the sides ofsimilar triangles?
358
358
208
1258
1258
208
4
3 3.5
8
67.5
328308
588
608
Skills Practice
Trigonometric Ratios
NAME ______________________________________________ DATE ____________ PERIOD _____
For each triangle, find sin T, cos T, and tan T to the nearest ten thousandth.
1. 2. 3.
Use a calculator to find the value of each trigonometric ratio to the nearest ten
thousandth.
4. sin 85° 5. cos 5° 6. tan 32.5°
Use a calculator to find the measure of each angle to the nearest degree.
7. sin D 5 0.5000 8. cos Q 5 0.1123 9. tan B 5 4.7465
For each triangle, find the measure of the indicated angle to the nearest degree.
10. 11. 12.
Solve each right triangle. State the side lengths to the nearest tenth and the angle
measures to the nearest degree.
13. 14. 15.
HIKING For Exercises 16 and 17, use the following information.
The 10-mile Lower West Rim trail in Mt. Zion National Park ascends 2640 feet.
16. What is the average angle of elevation of the hike from the canyon floor to the top of thecanyon? (Hint: Draw a diagram in which the length of the trail forms the hypotenuse ofa right triangle. Convert feet to miles.)
17. What is the horizontal distance covered on the hike?
18. RAMP DESIGN An engineer designed the entrance to a museum to include a wheelchairramp that is 15 feet long and forms a 6° angle with a sidewalk in front of the museum.How high does the ramp rise?
15 f30 ft
B C
A
13 cm648
A
C B
26 yd
518C B
A
1612
?40
42 ?13
9
?
16
88Ï·5
C
T
B25
724
R
P
T24
70
74T
X
Y
Practice
Trigonometric Ratios
NAME ______________________________________________ DATE ____________ PERIOD _____
11-711-7
Reading to Learn Mathematics
Trigonometric Ratios
NAME ______________________________________________ DATE ____________ PERIOD _____