During: The Pythagorean Theorem and Its converse

Transcription

1 Before: November 1st As a warm-up, let's do the Challenge Problems from the Quiz Yesterday 1. In Triangle ABC, centroid D is on median AM. AD = x - 3 and DM = 3x - 6. Find AM. 2. In Triangle ABC, centroid D is on median AM. AD = x - 3 and DM = 2x - 6. Find AM. Oct 30 9:08 PM During: The Pythagorean Theorem and Its converse Learning Target: To use the Pythagorean Theorem and its converse. 1

2 There is an important relationship in right triangles called the Pythagorean Theorem. This theorem is named for Pythagoras, a Greek mathematician who lived in the 500s B.C. We now know that the Babylonians, Egyptians, and Chinese were aware of this relationship before its discovery by Pythagoras. Theorem 8 1 2

3 Common Pythagorean triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 Problem 1: Finding the Length of the Hypotenuse I Do What is the length of the hypotenuse of triangle ABC? Do the side lengths of triangle ABC form a Pythagorean triple? Explain. 3

4 Problem 1: Finding the Length of the Hypotenuse We Do The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? Do the side lengths in part (a) form a Pythagorean triple? Problem 1: Finding the Length of the Hypotenuse You Do What is the length of the hypotenuse of triangle LMN? Do the side lengths of triangle LMN form a Pythagorean triple? Explain. 4

5 Problem 2: Finding the Length of a Leg I Do What is the value of x? Express your answer in simplest radical form. Problem 2: Finding the Length of a Leg We Do The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form. 5

6 Problem 2: Finding the Length of a Leg You Do What is the value of x? Express your answer in simplest radical form. Problem 3: Finding Distance I Do Dog agility courses often contain a seesaw obstacle, as shown below. To the nearest inch, how far above the ground are the dog s paws when the seesaw is parallel to the ground? 6

7 Problem 3: Finding Distance We Do The size of a computer monitor is the length of its diagonal. You want to buy a 19-in. monitor that has a height of 11 in. What is the width of the monitor? Round to the nearest tenth of an inch. Problem 3: Finding Distance You Do Jamal leans a 12-ft-long ladder against the side of a house. The base of the ladder is 4 ft. from the house. To the nearest tenth of a foot, how high on the house does the ladder reach? 7

8 You can use the Converse of the Pythagorean Theorem to determine whether a triangle is a right triangle. Theorem 8 2 Converse of the Pythagorean Theorem If Then... If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Triangle ABC is a right triangle. Problem 4: Identifying a Right Triangle I Do A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain. 8

9 Problem 4: Identifying a Right Triangle We Do A triangle has side lengths 16, 48, and 50. Is the triangle a right triangle? Explain. Once you know which length represents the hypotenuse, does it matter which length you substitute for a and which length you substitute for b? Explain. Problem 4: Identifying a Right Triangle You Do A triangle has side lengths 12.5, 30, and Is the triangle a right triangle? Explain. 9

10 Theorem 8 3 If Then If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. Triangle ABC is obtuse Theorem 8 4 If Then If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. Triangle ABC is acute Problem 5: Classifying a Triangle I Do A triangle has side lengths 6, 11, and 14. Is it acute, obtuse, or right? 10

11 Problem 5: Classifying a Triangle We Do Is a triangle with side lengths 7, 8, and 9 acute, obtuse, or right? Problem 5: Classifying a Triangle You Do A triangle has side lengths 4, 9, and 12. Is it acute, obtuse, or right? Explain. 11

12 After: Lesson Check What is the value of x in simplest radical form? After: Lesson Check Describe the conditions that a set of three numbers must meet in order to form a Pythagorean triple. A triangle has side lengths 16, 34, and 30. Your friend says it is not a right triangle. Look at your friend s work and describe the error. 12

13 Homework: Page 495, #7 21 odd, #22, #24 32 all Nov 2 8:58 AM 13