Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240
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Lesson 9-1 Squares and Square Roots Lesson 9-2 The Real Number System Lesson 9-3 Angles Lesson 9-4 Triangles Lesson 9-5 The Pythagorean Theorem Lesson 9-6 The Distance and Midpoint Formulas Lesson 9-7 Similar Triangles and Indirect Measurement Lesson 9-8 Sine, Cosine, and Tangent Ratios
Example 1 Find Square Roots Example 2 Calculate Square Roots Example 3 Estimate Square Roots Example 4 Use Square Roots to Solve a Problem
Find. indicates the positive square root of 64. Since Answer: 8
Find. indicates the negative square root of 121. Since Answer: 11
Find. indicates both square roots of 4. Since Answer: 2 and 2
Find each square root. a. b. Answer: 5 Answer: 12 c. Answer: 4 and 4
Use a calculator to find nearest tenth. to the 2nd [ ] 23 ENTER 4.79583152 Use a calculator. Round to the nearest tenth. Answer: 4.8 Check Since, the answer is reasonable.
Use a calculator to find nearest tenth. to the 2nd [ ] 46 ENTER 6.78232998 Use a calculator. Round to the nearest tenth. Answer: 6.8 Check Since, the answer is reasonable.
Use a calculator to find each square root to the nearest tenth. a. Answer: 8.4 b. Answer: 6.2
Estimate to the nearest whole number. Find the two perfect squares closest to 22. To do this, list some perfect squares. 1, 4, 9, 16, 25, 16 and 25 are closest to 22.
16 < 22 < 25 22 is between 16 and 25. < < 4 < < 5 is between and. and. Since 22 is closer to 25 than 16, the best whole number estimate for is 5. Answer: 5
Estimate to the nearest whole number. Find the two perfect squares closest to 319. To do this, list some perfect squares...., 225, 256, 289, 324, 289 and 324 are closest to 319.
324 < 319 < 289 319 is between 324 and 289. < < is between and. 18 < < 17 and.
Since 319 is closer to 324 than 289, the best whole number estimate for is 18. Check Answer: 18
Estimate each square root to the nearest whole number. a. Answer: 7 b. Answer: 12
Landmarks The observation deck at the Seattle Space Needle is 520 feet above the ground. On a clear day, about how far could a tourist on the deck see? Round to the nearest tenth. Use the formula where D is the distance in miles and A is the altitude, or height, in feet. Write the formula. Replace A with 520. Evaluate the square root first.
Multiply. Answer: On a clear day, a tourist could see about 27.8 miles.
Skyscraper A skyscraper stands 378 feet high. On a clear day, about how far could an individual standing on the roof of the skyscraper see? Round to the nearest tenth. Answer: On a clear day, an individual could see about 23.7 miles.
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Example 1 Classify Real Numbers Example 2 Compare Real Numbers on a Number Line Example 3 Solve Equations
Name all of the sets of numbers to which the real number 17 belongs. Answer: This number is a natural number, a whole number, an integer, and a rational number.
Name all of the sets of numbers to which the real number belongs. Answer: Since, this number is an integer and a rational number.
Name all of the sets of numbers to which the real number belongs. Answer: Since, this number is a natural number, a whole number, an integer, and a rational number.
Name all of the sets of numbers to which the real number belongs. Answer: This repeating decimal is a rational number because it is equivalent to.
Name all of the sets of numbers to which the real number belongs. Answer: It is not the square root of a perfect square so it is irrational.
Name all of the sets of numbers to which each real number belongs. a. 31 b. Answer: natural number, whole number, integer, rational number Answer: integer, rational number c. 4.375 d. e. Answer: rational number Answer: natural number, whole number, integer, rational number Answer: irrational number
Replace with <, >, or = to make true statement. a Express each number as a decimal. Then graph the numbers.
Answer: Since is to the left of
Order from least to greatest. Express each number as a decimal. Then compare the decimals.
Answer: From least to greatest, the order is
a. Replace with <, >, or = to make a true statement. Answer: > b. Order from least to greatest. Answer:
Solve if necessary.. Round to the nearest tenth, Write the equation. Take the square root of each side. Find the positive and negative square root. Answer: The solutions are 13 and 13.
Solve if necessary.. Round to the nearest tenth, Write the equation. Answer: The solutions are 7.1 and 7.1. Take the square root of each side. Find the positive and negative square root. Use a calculator.
Solve each equation. Round to the nearest tenth, if necessary. a. b. Answer: 9 and 9 Answer: 4.9 and 4.9
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Example 1 Measure Angles Example 2 Draw Angles Example 3 Classify Angles Example 4 Use Angles to Solve a Problem
Use a protractor to measure RSW. Step 1 Place the center point of the protractor s base on vertex S. Align the straight edge with side so that the marker for 0 is on the ray.
Use a protractor to measure RSW. 42 Step 2 Use the scale that begins with 0 at where the other side of the angle,, crosses this scale.. Read
Use a protractor to measure RSW. 42 Answer: The measure of angle RSW is 42. Using symbols,
Find the measurements of GUM, SUM, and BUG. 120 Answer: is at 0 on the right.
Find the measurements of GUM, SUM, and BUG. 32 Answer: is at 0 on the right.
Find the measurements of GUM, SUM, and BUG. 60 Answer: is at 0 on the left.
a. Use a protractor to measure ABC. Answer: 75
b. Find the measures of FDE, GDE, and HDG. Answer: FDE = 37, GDE = 118, HDG = 62
Draw R having a measurement of 145. R Step 1 Draw a ray with endpoint R.
Draw R having a measurement of 145. Step 2 Place the center point of the protractor on R. Align the mark labeled 0 with the ray. R
Draw R having a measurement of 145. 145 R Step 3 Use the scale that begins with 0. Locate the mark labeled 145. Then draw the other side of the angle.
Answer: 145 R
Draw M having a measurement of 47. Answer:
Classify the angle as acute, obtuse, right, or straight. m KLM < 90. Answer: KLM is acute.
Classify the angle as acute, obtuse, right, or straight. m NPQ = 180. Answer: NPQ is straight.
Classify the angle as acute, obtuse, right, or straight. m RST > 90. Answer: RST is obtuse.
Classify each angle as acute, obtuse, right, or straight. a. Answer: right b. Answer: obtuse
Classify each angle as acute, obtuse, right, or straight. c. Answer: straight
The diagram shows the angle between the back of a chair and the seat of the chair. Classify this angle. Answer: Since 95 is greater than 90, the angle is obtuse.
The diagram shows the angle between the bed of the truck and the frame of the truck. Classify this angle. Answer: The angle is acute.
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Example 1 Find Angle Measures Example 2 Use Ratios to Find Angle Measures Example 3 Classify Triangles
Find the value of x in DEF. The sum of the measures is 180. Replace m D with 100 and m E with 33. Simplify.
Answer: The measure of F is 47. Subtract 133 from each side.
Find the value of x in MNO. Answer: The measure of N is 57.
Algebra The measures of the angles of a certain triangle are in the ratio 2:3:5. What are the measures of the angles? Words Variables Equation The measures of the angles are in the ratio 2:3:5. Let 2x represent the measure of one angle, 3x the measure of a second angle, and 5x the measure of the third angle. The sum of the measures is 180.
Combine like terms. Divide each side by 10. Simplify. Since Answer: The measures of the angles are 36, 54, and 90.
Check So, the answer is correct.
Algebra The measures of the angles of a certain triangle are in the ratio 3:5:7. What are the measures of the angles? Answer: The measures of the angles are 36, 60, and 84.
Classify the triangle by its angles and by its sides. Angles Sides All angles are acute. All sides are congruent. Answer: The triangle is an acute equilateral triangle.
Classify the triangle by its angles and by its sides. Angles Sides The triangle has a right angle. The triangle has no congruent sides. Answer: The triangle is a right scalene triangle.
Classify each triangle by its angles and by its sides. a. Answer: obtuse scalene b. Answer: acute equilateral
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Example 1 Find the Length of the Hypotenuse Example 2 Solve a Right Triangle Example 3 Use the Pythagorean Theorem Example 4 Identify a Right Triangle
Find the length of the hypotenuse of the right triangle. Pythagorean Theorem Replace a with 21 and b with 20. Evaluate 21 2 and 20 2. Add 441 and 400.
Take the square root of each side. Answer: The length of the hypotenuse is 29 feet.
Find the length of the hypotenuse of the right triangle. Answer: The length of the hypotenuse is 5 meters.
Find the length of the leg of the right triangle. Pythagorean Theorem Replace c with 11 and a with 8. Evaluate 11 2 and 8 2.
Subtract 64 from each side. Simplify. Take the square root of each side. 2nd [ ] 57 ENTER 7.549834435 Answer: The length of the leg is about 7.5 meters.
Find the length of the leg of the right triangle. Answer: The length of the leg is about 12.7 inches.
Multiple-Choice Test Item A building is 10 feet tall. A ladder is positioned against the building so that the base of the ladder is 3 feet from the building. How long is the ladder? A 12.4 feet C 10.0 feet B 10.4 feet D 14.9 feet Read the Test Item Make a drawing to illustrate the problem. The ladder, ground, and side of the house form a right triangle.
Solve the Test Item Use the Pythagorean Theorem to find the length of the ladder. Pythagorean Theorem Replace a with 3 and b with 10. Evaluate 3 2 and 10 2. Simplify.
Take the square root of each side. Round to the nearest tenth. The ladder is about 10.4 feet tall. Answer: The answer is B.
Multiple-Choice Test Item An 18-foot ladder is placed against a building which is 14 feet tall. About how far is the base of the ladder from the building? A 11.6 feet C 11.3 feet B 10.9 feet D 11.1 feet Answer: The answer is C.
The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 48 ft, 60 ft, 78 ft The triangle is not a right triangle. Answer: no Pythagorean Theorem Replace a with 48, b with 60, and c with 78. Evaluate. Simplify.
The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 24 cm, 70 cm, 74 cm The triangle is a right triangle. Answer: yes Pythagorean Theorem Replace a with 24, b with 70, and c with 74. Evaluate. Simplify.
The measures of three sides of a triangle are given. Determine whether each triangle is a right triangle. a. 42 in., 61 in., 84 in. Answer: no b. 16 m, 30 m, 34 m Answer: yes
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Example 1 Use the Distance Formula Example 2 Use the Distance Formula to Solve a Problem Example 3 Use the Midpoint Formula
Find the distance between M(8, 4) and N( 6, 2). Round to the nearest tenth, if necessary. Use the Distance Formula. Distance Formula Simplify.
Evaluate ( 14) 2 and ( 6) 2. Add 196 and 36. Take the square root. Answer: The distance between points M and N is about 15.2 units.
Find the distance between A( 4, 5) and B(3, 9). Round to the nearest tenth, if necessary. Answer: The distance between points A and B is about 15.7 units.
Geometry Find the perimeter of XYZ to the nearest tenth. First, use the Distance Formula to find the length of each side of the triangle.
Distance Formula Simplify. Evaluate powers. Simplify.
Distance Formula Simplify. Evaluate powers. Simplify.
Distance Formula Simplify. Evaluate powers. Simplify.
Then add the lengths of the sides to find the perimeter. Answer: The perimeter is about 15.8 units.
Geometry Find the perimeter of ABC to the nearest tenth. Answer: The perimeter is about 21.3 units.
Find the coordinates of the midpoint of
Midpoint Formula Substitution Simplify. Answer: The coordinates of the midpoint of are (3, 3).
Find the coordinates of the midpoint of Answer: The coordinates of the midpoint of are (1, 1).
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Example 1 Find Measures of Similar Triangles Example 2 Use Indirect Measurement Example 3 Use Shadow Reckoning
If RUN ~ CAB, what is the value of x? The corresponding sides are proportional. Write a proportion.
Replace UR with 4, AC with 8, UN with 10, and AB with x. Find the cross products. Simplify. Mentally divide each side by 4. Answer: The value of x is 20.
If ABC ~ DEF, what is the value of x? Answer: The value of x is 3.
Maps A surveyor wants to find the distance RS across the lake. He constructs PQT similar to PRS and measures the distances as shown. What is the distance across the lake?
Write a proportion. Substitution Find the cross products. Simplify. Divide each side by 25. Answer: The distance across the lake is 28.8 meters.
Maps In the figure, MNO is similar to OPQ. Find the distance across the park. Answer: The distance across the park is 4.8 miles.
Landmarks Suppose the John Hancock Center in Chicago, Illinois, casts a 257.5-foot shadow at the same time a nearby tourist casts a 1.5-foot shadow. If the tourist is 6 feet tall, how tall is the John Hancock Center? Explore Plan Solve You know the lengths of the shadows and the height of the tourist. You need to find the height of the John Hancock Center. tourist s shadow Write and solve a proportion. tourist s height building s shadow building s height
Find the cross products. Multiply. Divide each side by 1.5. Answer: The height of the John Hancock Center is 1030 feet.
Building A man standing near a building casts a 2.5-foot shadow at the same time the building casts a 200-foot shadow. If the man is 6 feet tall, how tall is the building? Answer: The height of the building is 480 feet.
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Example 1 Find Trigonometric Ratios Example 2 Use a Calculator to Find Trigonometric Ratios Example 3 Use Trigonometric Ratios Example 4 Use Trigonometric Ratios to Solve a Problem
Find sin A, cos A, and tan A. Answer:
Find sin A, cos A, and tan A. Answer:
Find sin A, cos A, and tan A. Answer:
Find sin B, cos B, and tan B. Answer: sin B = 0.8; cos B = 0.6; tan B = 1.3333
Find the value of sin 19 to the nearest ten thousandth. SIN 19 ENTER 0.3255681545 Answer: sin 19 is about 0.3256.
Find the value of cos 51 to the nearest ten thousandth. COS 51 ENTER 0.629320391 Answer: cos 51 is about 0.6293.
Find the value of tan 24 to the nearest ten thousandth. TAN 24 ENTER 0.4452286853 Answer: tan 24 is about 0.4452.
Find each value to the nearest ten thousandth. a. sin 63 b. cos 14 c. tan 41 Answer: 0.8910 Answer: 0.9703 Answer: 0.8693
Find the missing measure. Round to the nearest tenth. The measures of an acute angle and the side adjacent to it are known. You need to find the measure of the hypotenuse. Use the cosine ratio. Write the cosine ratio.
Substitution Multiply each side by x. Simplify. Divide each side by cos 71.
12 COS 71 ENTER 36.85864184 Simplify. Answer: The measure of the hypotenuse is about 36.9 units.
Find the missing measure. Round to the nearest tenth. Answer: The measure of the missing side is about 21.4 units.
Architecture A tourist visiting the Petronas Towers in Kuala Lumpur, Malaysia, stands 261 feet away from their base. She looks at the top at an angle of 80 with the ground. How tall are the Towers? Use the tangent ratio. Write the tangent ratio.
Substitution Multiply each side by 261. 261 X TAN 80 ENTER 1480.204555 Simplify. Answer: The height of the Towers is about 1480.2 feet.
Architecture Jenna stands 142 feet away from the base of a building. She looks at the top at an angle of 62 with the ground. How tall is the building? Answer: The building is about 267.1 feet tall.
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