11.4 Apply the Pythagorean
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1 11.4 Apply the Pythagorean Theorem and its Converse Goal p and its converse. Your Notes VOCABULARY Hypotenuse Legs of a right triangle Pythagorean theorem THE PYTHAGOREAN THEOREM Words If a triangle is a right triangle, then the equals the. Algebra a b c Example 1 The lengths of the legs of a right triangle are a 5 8 and b Find c. c 2 5 a 2 1 b 2 Pythagorean theorem c Substitute for a and for b. c 2 5 Simplify. c 5 Take positive square root of each side. The side length of c is. Copyright Holt McDougal. All rights reserved. Lesson 11.4 Algebra 1 Notetaking Guide 295
2 11.4 Apply the Pythagorean Theorem and its Converse Goal p and its converse. Your Notes VOCABULARY Hypotenuse The hypotenuse of a right triangle is the side opposite the right angle. Legs of a right triangle The two sides that form the right angle Pythagorean theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. THE PYTHAGOREAN THEOREM Words If a triangle is a right triangle, then the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Algebra a 2 1 b 2 5 c 2 a b c Example 1 The lengths of the legs of a right triangle are a 5 8 and b Find c. c 2 5 a 2 1 b 2 Pythagorean theorem c Substitute 8 for a and 15 for b. c Simplify. c 5 17 Take positive square root of each side. The side length of c is 17. Copyright Holt McDougal. All rights reserved. Lesson 11.4 Algebra 1 Notetaking Guide 295
3 Checkpoint Complete the following exercises. 1. The lengths of the legs of a right triangle are a 5 7 and b 5 9. Find c. 2. The length of a leg of a right triangle is a 5 20 and the length of the hypotenuse is c Find b. Example 2 A right triangle has one leg that is 4 inches longer than the other leg. The hypotenuse is Ï } 106 inches. Find the unknown lengths. Sketch a right triangle and label the sides with their lengths. Let x be the length of the shorter leg. a 2 1 b 2 5 c 2 Pythagorean theorem 2 1 ( ) 2 5 ( ) 2 Substitute. 1 5 Simplify. 5 0 Write in standard form. 5 0 Factor. ( ) 5 0 or ( ) 5 0 Zero-product property x 5 or x 5 Solve for x. Because length is nonnegative, the solution x 5 does not make sense. The legs have lengths of inches and inches. 296 Lesson 11.4 Algebra 1 Notetaking Guide Copyright Holt McDougal. All rights reserved.
4 Checkpoint Complete the following exercises. 1. The lengths of the legs of a right triangle are a 5 7 and b 5 9. Find c. c 5 Ï } The length of a leg of a right triangle is a 5 20 and the length of the hypotenuse is c Find b. b 5 48 Example 2 A right triangle has one leg that is 4 inches longer than the other leg. The hypotenuse is Ï } 106 inches. Find the unknown lengths. Sketch a right triangle and label the sides with their lengths. Let x be the length of the shorter leg. a 2 1 b 2 5 c 2 x 2 1 ( x 1 4 ) 2 5 ( Ï } 106 ) 2 x 2 1 x 2 1 8x x 2 1 8x x x Pythagorean theorem Substitute. Simplify. Write in standard form. 2(x 1 9)(x 2 5) 5 0 Factor. ( x 1 9 ) 5 0 or ( x 2 5 ) 5 0 Zero-product property x 5 29 or x 5 5 Solve for x. Because length is nonnegative, the solution x 5 29 does not make sense. The legs have lengths of 5 inches and inches. 296 Lesson 11.4 Algebra 1 Notetaking Guide Copyright Holt McDougal. All rights reserved.
5 Checkpoint Complete the following exercise. 3. A right triangle has one leg that is 2 centimeters shorter than the other leg. The length of the hypotenuse is 10 centimeters. Find the unknown lengths. CONVERSE OF THE PYTHAGOREAN THEOREM If a triangle has side lengths a, b, and c such that, then the triangle is a triangle. Example 3 Determine right triangles Tell whether the triangle with the given side lengths is a a. 10, 11, 15 b. 3, 4, The triangle a The triangle a Checkpoint Tell whether the triangle with the given side lengths is a 4. 9, 40, , 15, 18 Homework 6. A triangular mirror has side lengths of 1.2 meters, 1.6 meters, and 2 meters. Is the mirror a right triangle? Explain. Copyright Holt McDougal. All rights reserved. Lesson 11.4 Algebra 1 Notetaking Guide 297
6 Checkpoint Complete the following exercise. 3. A right triangle has one leg that is 2 centimeters shorter than the other leg. The length of the hypotenuse is 10 centimeters. Find the unknown lengths. 6 cm, 8 cm CONVERSE OF THE PYTHAGOREAN THEOREM If a triangle has side lengths a, b, and c such that a 2 1 b 2 5 c 2, then the triangle is a Example 3 Determine right triangles Tell whether the triangle with the given side lengths is a a. 10, 11, 15 b. 3, 4, Þ The triangle is not a The triangle is a Checkpoint Tell whether the triangle with the given side lengths is a 4. 9, 40, , 15, 18 The triangle is a The triangle is not a Homework 6. A triangular mirror has side lengths of 1.2 meters, 1.6 meters, and 2 meters. Is the mirror a right triangle? Explain. Yes, The sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Copyright Holt McDougal. All rights reserved. Lesson 11.4 Algebra 1 Notetaking Guide 297
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To change from tonnes to kilograms, multiply by This means: tonnes = 225 kg [ = 225]
![To change from tonnes to kilograms, multiply by This means: tonnes = 225 kg [ = 225]](/thumbs/96/127347354.jpg "To change from tonnes to kilograms, multiply by This means: tonnes = 225 kg [ = 225]")
Q1. [1 tonne = 1000 kg] To change from tonnes to kilograms, multiply by 1000. This means: 0.225 tonnes = 225 kg [0.225 1000 = 225] Difference between 0.225 tonnes and 128 kg means the same as: Difference
The study of the measurement of triangles is called Trigonometry.
Math 10 Workplace & Apprenticeship 7.2 The Sine Ratio Day 1 Plumbers often use a formula to determine the lengths of pipes that have to be fitted around objects. Some common terms are offset, run, and
Use SOH CAH TOA to memorize the three main trigonometric functions.
Use SOH CAH TOA to memorize the three main trigonometric functions. Content Objective Content Objective Content Objective Content Objective Content Objective Content Objective Content Objective Content
8-5 Angles of Elevation and Depression
4. HOCKEY A hockey player takes a shot 20 feet away from a 5-foot goal. If the puck travels at a angle of elevation toward the center of the goal, will the player score? 5. MOUNTAINS Find the angle of