Pythagorean Triples the Converse of the Theorem Lesson

Pythagorean Triples & the Converse of the Theorem Lesson 3. 3. 4

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem California Standard: What it means for you: Measurement and Geometry 3. 3 You’ll learn about the groups of whole numbers that make the Pythagorean theorem true, and how to use the converse of theorem to find out if a triangle is a right triangle. Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Key words: • • • Pythagorean theorem Pythagorean triple converse right angle acute obtuse 2

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Up to now you’ve been using the Pythagorean theorem on triangles that you’ve been told are right triangles. But if you don’t know for sure whether a triangle is a right triangle, you can use theorem to decide. It’s kind of like using theorem backwards — and it’s called using the converse of theorem. 3

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Pythagorean Triples are All Whole Numbers You can draw a right triangle with any length legs you like, so the list of side lengths that can make the equation c 2 = a 2 + b 2 true never ends. Most sets of side lengths that fit the equation include at least one decimal — that’s because finding the length of the hypotenuse using the equation involves taking a square root. 4

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem There are some sets of side lengths that are all integers — these are called Pythagorean triples. You’ve seen a lot of these already. For example: (3, 4, 5) (6, 8, 10) (8, 15, 17) (5, 12, 13) 5 4 17 3 15 13 10 8 6 12 8 5 5

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem You can find more Pythagorean triples by multiplying each of the numbers in a triple by the same number. For example: (3, 4, 5) × 2 × 3 × 4 (6, 8, 10) (9, 12, 15) These are all Pythagorean triples. (12, 16, 20) To test if three integers are a Pythagorean triple, put them into the equation c 2 = a 2 + b 2, where c is the biggest of the numbers. If they make the equation true, they’re a Pythagorean triple. If they don’t, they’re not. 6

Lesson 3. 3. 4 Example Pythagorean Triples & the Converse of the Theorem 1 Are the numbers (72, 96, 120) a Pythagorean triple? Solution To see if the numbers are a Pythagorean triple, put them into the equation. c 2 = a 2 + b 2 Write out the equation 1202 = 722 + 962 Substitute the values given 14, 400 = 5184 + 9216 Simplify the equation 14, 400 = 14, 400 — which is true These numbers are a Pythagorean triple. 7 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Guided Practice Are the sets of numbers in Exercises 1– 6 Pythagorean triples or not? If they are not, give a reason why not. 1. 5, 12, 13 2. 0. 7, 0. 9, 1. 4 Pythagorean triple 3. 8, 8, No, they’re not integers and they don’t fit the equation. 4. 25, 60, 65 No, they’re not all integers. 5. 6, 9, 12 Pythagorean triple 6. 18, 80, 82 No, they don’t fit the equation. Pythagorean triple 8 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem The Converse of the Pythagorean Theorem The Pythagorean theorem says that the side lengths of any right triangle will satisfy the equation c 2 = a 2 + b 2, where c is the hypotenuse and a and b are the leg lengths. You can also say the opposite — if a triangle’s side lengths satisfy the equation, it is a right triangle. This is called the converse of theorem: The converse of the Pythagorean theorem: If the side lengths of a triangle a, b, and c, where c is the largest, satisfy the equation c 2 = a 2 + b 2, then the triangle is a right triangle. 9

Lesson 3. 3. 4 Example Pythagorean Triples & the Converse of the Theorem 2 A triangle has side lengths 2. 5 cm, 6 cm, and 6. 5 cm. Is it a right triangle? Solution Put the side lengths into the equation c 2 = a 2 + b 2, and evaluate both sides. c 2 = a 2 + b 2 Write out the equation 6. 52 = 2. 52 + 62 Substitute the values given 42. 25 6. 25 side + 36 of a right Simplify the equation The = longest triangle is the hypotenuse, c. 42. 25 = 42. 25 — which is true It is a right triangle. 10 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Test Whether a Triangle is Acute or Obtuse If a triangle isn’t a right triangle, it must either be an acute triangle or an obtuse triangle. Acute — all angles < 90° Obtuse — one angle between 90° and 180° By seeing if c 2 is greater than or less than a 2 + b 2, you can tell what type of triangle it is. If c 2 > a 2 + b 2 then the triangle is obtuse. If c 2 < a 2 + b 2 then the triangle is acute. 11

Lesson 3. 3. 4 Example Pythagorean Triples & the Converse of the Theorem 3 A triangle has side lengths of 2 ft, 2. 5 ft, and 3 ft. Is it right, acute, or obtuse? Solution Check whether c 2 = a 2 + b 2, with c = 3, a = 2 and b = 2. 5. c 2 = 3 2 = 9 Find the value of c 2 a 2 + b 2 = 22 + 2. 52 = 4 + 6. 25 = 10. 25 9 < 10. 25 Now find a 2 + b 2 Compare the two values c 2 < a 2 + b 2, so this is an acute triangle. 12 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Guided Practice Are the side lengths in Exercises 7– 12 of right, acute, or obtuse triangles? 7. 50, 120, 130 8. 8, 9, 10 Right triangle 9. 3, 4, 6 Acute triangle 10. 12, 6, Obtuse triangle 11. 0. 25, 0. 3, 0. 5 Obtuse triangle Right triangle 12. 0. 5, 0. 52, 0. 55 Acute triangle 13 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Independent Practice 1. “Every set of numbers that satisfies the equation c 2 = a 2 + b 2 is a Pythagorean triple. ” Explain if this statement is true or not. Not true: a Pythagorean triple must be made up of positive integers. 14 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Independent Practice Say if the side lengths in Exercises 2– 7 are Pythagorean triples or not. 2. 8, 15, 17 Pythagorean triple 4. 0. 3, 0. 4, 0. 5 Not 6. 12, 29, 40 Not 3. 1, 1, Not 5. 300, 400, 500 Pythagorean triple 7. 15, 36, 39 Pythagorean triple 15 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Independent Practice 8. In triangle ABC, side AB is longest. If AB 2 > AC 2 + BC 2 then what kind of triangle is ABC? Triangle ABC is obtuse. 16 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Independent Practice Are the following side lengths those of right, acute, or obtuse triangles? 9. 5, 10, 14 Obtuse triangle 11. 12, 16, 20 Right triangle 13. 2. 4, 4. 5, 5. 1 Right triangle 10. 10, 11, 12 Acute triangle 12. 5. 1, 5. 3, 9. 3 Obtuse triangle 14. 3. 7, 4. 7, 5. 7 Acute triangle 17 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Independent Practice 15. Justin is going to fit a new door. He measures the width of the door frame as 105 cm, the height as 200 cm, and the diagonal of the frame as 232 cm. Is the door frame perfectly rectangular? No: its sides and diagonal form an obtuse triangle, not a right triangle. 18 Solution follows…

Lesson 3. 3. 4 Pythagorean Triples & the Converse of the Theorem Round Up The converse of the Pythagorean theorem is a kind of “backward” version. You can use it to prove whether a triangle is a right triangle or not — and if it’s not, you can say if it’s acute or obtuse. 19