Welcome to Interactive Chalkboard Algebra 1 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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Contents Lesson 11-1Simplifying Radical Expressions Lesson 11-2Operations with Radical Expressions Lesson 11-3Radical Equations Lesson 11-4The Pythagorean Theorem Lesson 11-5The Distance Formula Lesson 11-6Similar Triangles Lesson 11-7Trigonometric Ratios

Lesson 1 Contents Example 1Simplify Square Roots Example 2Multiply Square Roots Example 3Simplify a Square Root with Variables Example 4Rationalizing the Denominator Example 5Use Conjugates to Rationalize a Denominator

Example 1-1a Simplify. Prime factorization of 52 Product Property of Square Roots Answer: Simplify.

Example 1-1b Simplify. Product Property of Square Roots Simplify. Prime factorization of 72 Answer: Simplify.

Example 1-1c Simplify. a. b. Answer:

Example 1-2a Find Product Property Answer: Simplify. Product Property

Example 1-2b Find Answer:

Example 1-3a Simplify Prime factorization Product Property Simplify. The absolute value of ensures a nonnegative result. Answer:

Example 1-3b Simplify Answer:

Example 1-4a Simplify. Answer: Simplify. Product Property of Square Roots Multiply by.

Example 1-4b Simplify. Prime factorization Multiply by Answer: Product Property of Square Roots

Example 1-4c Product Property of Square Roots Prime factorization Simplify. Multiply by

Example 1-4c Answer: Divide the numerator and denominator by 2.

Example 1-4d Simplify. a. b. c. Answer:

Example 1-5a Simplify Simplify. Answer:

Example 1-5b Simplify Answer:

End of Lesson 1

Lesson 2 Contents Example 1Expressions with Like Radicands Example 2Expressions with Unlike Radicands Example 3Multiply Radical Expressions

Example 2-1a Simplify. Distributive Property Answer: Simplify.

Example 2-1b Simplify. Commutative Property Distributive Property Answer: Simplify.

Example 2-1c Simplify a. b. Answer:

Example 2-2a Simplify Answer: The simplified form is

Example 2-2b Simplify Answer:

Example 2-3a Find the area of a rectangle with a width of and a length of To find the area of the rectangle multiply the measures of the length and width.

Example 2-3a Answer: The area of the rectangle is square units. First termsOuter termsInner termsLast terms Multiply. Prime factorization Simplify. Combine like terms.

Example 2-3b Find the area of a rectangle with a width of and a length of Answer: units 2

End of Lesson 2

Lesson 3 Contents Example 1Radical Equation with a Variable Example 2Radical Equation with an Expression Example 3Variable on Each Side

Example 3-1a Free-Fall Height An object is dropped from an unknown height and reaches the ground in 5 seconds. From what height is it dropped? Use the equationand replace t with 5 seconds. Original equation Replace t with 5. Multiply each side by 4. Square each side. Simplify.

Example 3-1a Check Original equation Answer: The object is dropped from 400 feet. and

Example 3-1b Free-Fall Height An object is dropped from an unknown height and reaches the ground in 7 seconds. Use the equationto find from what height it is dropped. Answer: 784 ft

Example 3-2a Solve Answer: The solution is 52. Original equation Subtract 8 from each side. Square each side. Add 3 to each side.

Example 3-2b Answer: 60 Solve

Example 3-3a Solve Original equation Square each side. Simplify. Subtract 2 and add y to each side. Factor. Solve. Zero Product Property or

Example 3-3a Check Answer: Since –2 does not satisfy the original equation, 1 is the only solution.

Example 3-3b Solve Answer: 3

End of Lesson 3

Lesson 4 Contents Example 1Find the Length of the Hypotenuse Example 2Find the Length of a Side Example 3Pythagorean Triples Example 4Check for Right Triangles

Example 4-1a Find the length of the hypotenuse of a right triangle if and Answer: The length of the hypotenuse is 30 units. Pythagorean Theorem Simplify. Take the square root of each side. Use the positive value. and

Example 4-1b Find the length of the hypotenuse of a right triangle if and Answer: 65 units

Example 4-2a Find the length of the missing side. In the triangle,andunits. Answer: To the nearest hundredth, the length of the leg is units. Pythagorean Theorem Evaluate squares. Subtract 81 from each side. Use a calculator to evaluate. Use the positive value. and

Example 4-2b Find the length of the missing side. Answer: about units

Example 4-3a Multiple-Choice Test Item What is the area of triangle XYZ ? A 94 units 2 B 128 units 2 C 294 units 2 D 588 units 2 Read the Test Item The area of the triangle isIn a right triangle, the legs form the base and height of the triangle. Use the measures of the hypotenuse and the base to find the height of the triangle.

Example 4-3a Solve the Test Item Step 1Check to see if the measurements of this triangle are a multiple of a common Pythagorean triple. The hypotenuse isunits and the leg isunits. This triangle is a multiple of a (3, 4, 5) triangle. The height of the triangle is 21 units.

Example 4-3a Step 2Find the area of the triangle. Answer: The area of the triangle is 294 square units. Choice C is correct. Area of a triangle and Simplify.

Example 4-3b Multiple-Choice Test Item What is the area of triangle RST ? A 764 units 2 B 480 units 2 C 420 units 2 D 384 units 2 Answer: D

Example 4-4a Determine whether the side measures of 7, 12, 15 form a right triangle. Answer: Since, the triangle is not a right triangle. Pythagorean Theorem Add. and Multiply. Since the measure of the longest side is 15, let, and Then determine whether

Example 4-4b Determine whether the side measures of 27, 36, 45 form a right triangle. Pythagorean Theorem and Multiply. Since the measure of the longest side is 45, let and Then determine whether Add. Answer: Sincethe triangle is a right triangle.

Determine whether the following side measures form right triangles. a. 33, 44, 55 b. 12, 22, 40 Example 4-4b Answer: right triangle Answer: not a right triangle

End of Lesson 4

Lesson 5 Contents Example 1Distance Between Two Points Example 2Use the Distance Formula Example 3Find a Missing Coordinate

Example 5-1a Find the distance between the points at (1, 2) and (–3, 0). Answer:or about 4.47 units Distance Formula and Simplify. Evaluate squares and simplify.

Example 5-1b Find the distance between the points at (5, 4) and (0, –2). Answer:

Example 5-2a Biathlon Julianne is sighting her rifle for an upcoming biathlon competition. Her first shot is 2 inches to the right and 7 inches below the bull’s-eye. What is the distance between the bull’s-eye and where her first shot hit the target? Draw a model of the situation on a coordinate grid. If the bull’s-eye is at (0, 0), then the location of the first shot is (2, –7). Use the Distance Formula.

Example 5-2a

Distance Formula and Simplify. or about 7.28 inches Answer: The distance isor about 7.28 inches.

Example 5-2b Horseshoes Marcy is pitching a horseshoe in her local park. Her first pitch is 9 inches to the left and 3 inches below the pin. What is the distance between the horseshoe and the pin? Answer:

Example 5-3a Find the value of a if the distance between the points at (2, –1) and (a, –4) is 5 units. Distance Formula Simplify. Evaluate squares.Simplify. Let and.

Example 5-3a Square each side. Subtract 25 from each side. Factor. Solve. Answer: The value of a is –2 or 6. Zero Product Property or

Example 5-3b Find the value of a if the distance between the points at (2, 3) and (a, 2) is Answer: –4 or 8

End of Lesson 5

Lesson 6 Contents Example 1Determine Whether Two Triangles Are Similar Example 2Find Missing Measures Example 3Use Similar Triangles to Solve a Problem

Example 6-1a Determine whether the pair of triangles is similar. Justify your answer. The ratio of sides XY to AB is The ratio of sides YZ to BC is

Example 6-1a The ratio of sides XZ to AC is Answer: Since the measures of the corresponding sides are proportional, triangle XYZ is similar to triangle ABC.

Example 6-1b Determine whether the pair of triangles is similar. Justify your answer. Answer: Since the corresponding angles have equal measures, the triangles are similar.

Example 6-2a Find the missing measures if the pair of triangles is similar. Since the corresponding angles have equal measures, The lengths of the corresponding sides are proportional.

Example 6-2a Corresponding sides of similar triangles are proportional. and Find the cross products. Divide each side by 18.

Example 6-2a Corresponding sides of similar triangles are proportional. and Find the cross products. Divide each side by 18. Answer: The missing measures are 27 and 12.

Example 6-2b Find the missing measures if the pair of triangles is similar. Corresponding sides of similar triangles are proportional. and

Example 6-2b Answer: The missing measure is 7.5. Find the cross products. Divide each side by 4.

Example 6-2c Find the missing measures if each pair of triangles is similar. a. Answer: The missing measures are 18 and 42.

Example 6-2c Answer: The missing measure is 5.25 mm. Find the missing measures if each pair of triangles is similar. b. a

Example 6-3a Since the length of the shadow of the tree and Richard’s height are given in meters, convert the length of Richard’s shadow to meters. Shadows Richard is standing next to the General Sherman Giant Sequoia tree in Sequoia National Park. The shadow of the tree is 22.5 meters, and Richard’s shadow is 53.6 centimeters. If Richard’s height is 2 meters, how tall is the tree?

Example 6-3a Let the height of the tree. Simplify. Richard’s shadow Tree’s shadow Richard’s height Tree’s height Answer: The tree is about 84 meters tall. Cross products

Example 6-3b Answer: The length of Trudie’s shadow is about 0.98 meter. Tourism Trudie is standing next to the Eiffel Tower in France. The height of the Eiffel Tower is 317 meters and casts a shadow of 155 meters. If Trudie’s height is 2 meters, how long is her shadow?

End of Lesson 6

Lesson 7 Contents Example 1Sine, Cosine, and Tangent Example 2Find the Sine of an Angle Example 3Find the Measure of an Angle Example 4Solve a Triangle Example 5Angle of Elevation

Example 7-1a Find the sine, cosine, and tangent of each acute angle of Round to the nearest ten-thousandth. Write each ratio and substitute the measures. Use a calculator to find each value.

Example 7-1a Answers: Answer:

Example 7-1a Answers: Answer:

Find the sine, cosine, and tangent of each acute angle of Round to the nearest ten-thousandth. Example 7-1b Answer:

Example 7-2a Answer: Rounded to the nearest ten thousandth, Find cos 65° to the nearest ten thousandth. Keystrokes COS ENTER

Example 7-2b Answer: Find tan 32° to the nearest ten thousandth.

Example 7-3a and Since the lengths of the adjacent leg and the hypotenuse are known, use the cosine ratio. Now use [COS –1 ] on a calculator to find the measure of the angle whose cosine ratio is Find the measure of to the nearest degree.

Example 7-3a Answer: To the nearest degree, the measure ofis 53°. [COS –1 ] ENTER Keystrokes 20 2nd

Find the measure ofto the nearest degree. Example 7-3b Answer: 29°

Step 1 Find the measure ofThe sum of the measures of the angles in a triangle is 180. The measure of is 28°. Example 7-4a You need to find the measures of and Find all of the missing measures in

Definition of sine Evaluate sin 62°. Multiply each side by y and divide each side by is about 17.0 centimeters long. Step 2 Find the value of y, which is the measure of the hypotenuse. Since you know the measure of the side opposite, use the sine ratio. Example 7-4a

Step 3 Find the value of x, which is the measure of the side adjacent Use the tangent ratio. Find the cross products. is about 8.0 centimeters long. Answer: So, the missing measures are 28 , 8 cm, and 17 cm. Example 7-4a Definition of tangent tan 28º Evaluate tan 28°

Find all of the missing measures in Example 7-4b Answer: The missing measures are 47 , 11 m, and 16 m.

Example 7-5a ExploreIn the diagram two right triangles are formed. You know the height of the airplane and the horizontal distance it has traveled. PlanLet A represent the first angle of depression. Let B represent the second angle of depression. Indirect Measurement In the diagram, Barone is flying his model airplane 400 feet above him. An angle of depression is formed by a horizontal line of sight and a line of sight below it. Find the angles of depression at points A and B to the nearest degree.

and Example 7-5a SolveWrite two equations involving the tangent ratio. Answer: The angle of depression at point A is 45° and the angle of depression at point B is 37°.

Example 7-5a ExamineExamine the solution by finding the horizontal distance the airplane has flown at points A and B. The solution checks.

Example 7-5b Answer:The angle of depression at point X is 38° and the angle of depression at Y is 32°. Indirect Measurement In the diagram, Kylie is flying a kite 350 feet above her. An angle of depression is formed by a horizontal line of sight and a line of sight below it. Find the angle of depression at points X and Y to the nearest degree.

End of Lesson 7

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