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FeatureLesson Geometry Lesson Main (For help, go to the Skills Handbook, page 753.) 1.2. 3.4. Lesson 8-1 The Pythagorean Theorem and Its Converse Square.

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Presentation on theme: "FeatureLesson Geometry Lesson Main (For help, go to the Skills Handbook, page 753.) 1.2. 3.4. Lesson 8-1 The Pythagorean Theorem and Its Converse Square."— Presentation transcript:

1 FeatureLesson Geometry Lesson Main (For help, go to the Skills Handbook, page 753.) Lesson 8-1 The Pythagorean Theorem and Its Converse Square the lengths of the sides of each triangle. What do you notice? Check Skills Youll Need 8-1

2 FeatureLesson Geometry Lesson Main = (3)(3) = 9; 4 2 = (4)(4) = 16; 5 2 = (5)(5) = 25; = = (5)(5) = 25; 12 2 = (12)(12) = 144; 13 2 = (13)(13) = 169; = = (6)(6) = 36; 8 2 = (8)(8) = 64; 10 2 = (10)(10) = 100; = = (4)(4) = 16; (4 2) 2 = (4 2)(4 2) = 16 (2)(2) = 16(2) = 32; = 32. Solutions Lesson 8-1 The Pythagorean Theorem and Its Converse Check Skills Youll Need 8-1

3 FeatureLesson Geometry Lesson Main Lesson 8-1 The Pythagorean Theorem and Its Converse Notes 8-1

4 FeatureLesson Geometry Lesson Main Lesson 8-1 The Pythagorean Theorem and Its Converse Notes 8-1 A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a 2 + b 2 = c 2. Some common Pythagorean triples are: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

5 FeatureLesson Geometry Lesson Main Lesson 8-1 The Pythagorean Theorem and Its Converse Notes 8-1

6 FeatureLesson Geometry Lesson Main Lesson 8-1 The Pythagorean Theorem and Its Converse Notes 8-1

7 FeatureLesson Geometry Lesson Main Lesson 8-1 The Pythagorean Theorem and Its Converse Notes 8-1

8 FeatureLesson Geometry Lesson Main A right triangle has legs of length 16 and 30. Find the length of the hypotenuse. Do the lengths of the sides form a Pythagorean triple? a 2 + b 2 = c 2 Use the Pythagorean Theorem = c 2 Substitute 16 for a and 30 for b = c 2 Simplify = c 2 34 = cTake the square root. The length of the hypotenuse is 34. The lengths of the sides, 16, 30, and 34, form a Pythagorean triple because they are whole numbers that satisfy a 2 + b 2 = c 2. Notice that each length is twice the common Pythagorean triple of 8, 15, and 17. Lesson 8-1 The Pythagorean Theorem and Its Converse Quick Check Additional Examples 8-1 Pythagorean Triples

9 FeatureLesson Geometry Lesson Main a 2 + b 2 = c 2 Use the Pythagorean Theorem. x = 12 2 Substitute x for a, 10 for b, and 12 for c. x = 144 Simplify. x 2 = 44Subtract 100 from each side. x = 4(11) Take the square root of each side. x = 2 11 Simplify. Find the value of x. Leave your answer in simplest radical form. Lesson 8-1 The Pythagorean Theorem and Its Converse Quick Check Additional Examples 8-1 Using Simplest Radical Form

10 FeatureLesson Geometry Lesson Main c = 16,200 Take the square root. c Use a calculator. a 2 + b 2 = c 2 Use the Pythagorean Theorem = c 2 Substitute 90 for a and for b. 8, ,100 = c 2 Simplify. 16,200 = c 2 The distance to home plate from second base is about 127 ft. Use the information to draw a baseball diamond. A baseball diamond is a square with 90-ft sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base? Lesson 8-1 The Pythagorean Theorem and Its Converse Quick Check Additional Examples 8-1 Real-World Connection

11 FeatureLesson Geometry Lesson Main Is this triangle a right triangle? Lesson 8-1 The Pythagorean Theorem and Its Converse Because a 2 + b 2 c 2, the triangle is not a right triangle. Quick Check a 2 + b 2 c Substitute 4 for a, 6 for b, and 7 for c Simplify. Additional Examples 8-1 Is It a Right Triangle?

12 FeatureLesson Geometry Lesson Main The numbers represent the lengths of the sides of a triangle. Classify each triangle as acute, obtuse, or right. Lesson 8-1 The Pythagorean Theorem and Its Converse a.15, 20, 25 c 2 a 2 + b 2 Compare c 2 with a 2 + b Simplify. 625 = 625 Because c 2 = a 2 + b 2, the triangle is a right triangle Substitute the greatest length for c. Additional Examples 8-1 Classifying Triangles as Obtuse, Acute, or Right

13 FeatureLesson Geometry Lesson Main (Continued) Lesson 8-1 The Pythagorean Theorem and Its Converse b. 10, 15, Because c 2 a 2 + b 2, the triangle is obtuse. Quick Check Substitute the greatest length for c Simplify. c 2 a 2 + b 2 Compare c 2 with a 2 + b 2. Additional Examples 8-1

14 FeatureLesson Geometry Lesson Main Find the value of x. 2. Find the value of x. Leave your answer in simplest radical form. 3. The town of Elena is 24 mi north and 8 mi west of Holberg. A train runs on a straight track between the two towns. How many miles does it cover? Round your answer to the nearest whole number. 4. The lengths of the sides of a triangle are 5 cm, 8 cm, and 10 cm. Is it acute, right, or obtuse? mi obtuse Lesson 8-1 The Pythagorean Theorem and Its Converse Lesson Quiz 8-1


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