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Lessons 9.1 – 9.2 The Pythagorean Theorem & Its Converse

Lessons 9.1 – 9.2 The Pythagorean Theorem & Its Converse. HW: Lesson 9.1 / 1-16 evens and Lesson 9.2/1-16 evens. Essential Understanding. Use the the Pythagorean Theorem to solve problems. Use the Converse of the Pythagorean Theorem to solve problems.

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Lessons 9.1 – 9.2 The Pythagorean Theorem & Its Converse

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  1. Lessons 9.1 – 9.2 The Pythagorean Theorem & Its Converse HW: Lesson 9.1 / 1-16 evens and Lesson 9.2/1-16 evens

  2. Essential Understanding • Use the the Pythagorean Theorem to solve problems. • Use the Converse of the Pythagorean Theorem to solve problems. • Use side lengths to classify triangles by their angle measures.

  3. Pythagorean Theorem If You Have A Right Triangle, Then c²=a² + b² c a b

  4. c a b The Pythagorean Theorem as some students see it. c2 = a2 + b2

  5. A better way c2 c a2 a c2=a2+b2 b b2

  6. PYTHAGOREAN THEOREM Applies to Right Triangles Only! hypotenuse c leg a b leg c2=a2+b2

  7. 1 x 3 cm 4 cm 2 x 5 cm 12 cm Pythagoras Questions Pythagorean triple Pythagorean triple

  8. 3 11m x m 9 m 4 23.8 cm 11 cm x cm Pythagoras Questions: Finding a leg measure x ≈ 6.32 cm Another method for finding a leg measure x ≈ 21.11 cm

  9. Applications of Pythagoras 1 Find the diagonal of the rectangle 6 cm d 9.3 cm d = 11.07 cm

  10. A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm. Find its perimeter. 2 7.8 cm 4.3 cm x cm x ≈ 6.51 cm Perimeter = 2(6.51+4.3) ≈ 21.62 cm therefore

  11. The Converse Of The Pythagorean Theorem If c² =a² + b², Then You Have A Right Triangle c a b

  12. Do These Lengths Form Right Triangles? i.e. do they work in the Pythagorean Theorem? 5, 6, 10 6, 8, 10 10² __5² + 6² 100___25 + 36 100≠ 61 NO 10²___6² + 8² 100___36 + 64 100 = 100 YES

  13. Determine whether a triangle with lengths 7, 11, and 12 form a right triangle. **The hypotenuse is the longest length. Example of the Converse This is not a right triangle.

  14. A Pythagorean Triple Is Any 3 Integers That Form A Right Triangle 5, 12, 13 Multiples Family 10,24,26 25,60,65 35,84,91 3, 4, 5 Multiples Family 6,8,10 30,40,50 15,20,25 Multiples of Pythagorean Triples are also Pythagorean Triples.

  15. Determine whether a triangle with lengths 12, 20, and 16 form a right triangle. Example of the Converse This is a right triangle. A set of integers such as 12, 16, and 20 is a Pythagorean triple.

  16. Determine whether 4, 5, 6 is a Pythagorean triple. Determine whether 15, 8, and 17 is a Pythagorean triple. Converse Examples 4, 5, and 6 is not a Pythagorean triple. 15, 8, and 17 is a Pythagorean triple.

  17. Verifying Right Triangles ? ? The triangle is a right triangle. Note: squaring a square root!!

  18. Verifying Right Triangles ? ? ? The triangle is NOT a right triangle. Note: squaring an integer & square root!!

  19. What Kind of Triangle?? You can use the Converse of the Pythagorean Theorem to verify that a given triangle is a right triangle or obtuse or acute. What Kind Of Triangle ? c² ?? a² + b²

  20. Triangle Inequality What Kind Of Triangle ? c² ?? a² + b² If the c²= a² + b² , then right If the c²> a² + b² then obtuse If the c²< a² + b², then acute The converse of the Pythagorean Theorem can be used to categorize triangles.

  21. 38, 77, 86 c2? a2 + b2 862? 382 + 772 7396 ? 1444 + 5959 7396 > 7373 Triangle Inequality The triangle is obtuse

  22. 10.5, 36.5, 37.5 c2? a2 + b2 37.52? 10.52 + 36.52 1406.25 ? 110.25 + 1332.25 1406.24 < 1442.5 Triangle Inequality The triangle is acute

  23. 4,7,9 9²__4² + 7² 81__16 + 49 81 > 65 OBTUSE greater

  24. 5,5,7 • 7² __5² + 5² • __ 25 +25 • 49 < 50 • ACUTE Less than

  25. A Pythagorean Triple 3, 4, 5 25 9 52=32+ 42 5 3 4 16 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 25=9 + 16

  26. A 2nd Pythagorean Triple 5, 12, 13 169 13 25 5 12 132 =52 + 122 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 144 169=25 + 144

  27. A 3rd Pythagorean Triple 625 7, 24, 25 49 25 7 24 252 =72+ 242 576 625=49 + 576

  28. Construction: You use four stakes and string to mark the foundation of a house. You want to make sure the foundation is rectangular. a. A friend measures the four sides to be 30 feet, 30 feet, 72 feet, and 72 feet. He says these measurements prove that the foundation is rectangular. Is he correct? Building a foundation

  29. Building a foundation • Solution: Your friend is not correct. The foundation could be a nonrectangular parallelogram, as shown below.

  30. Building a foundation b. You measure one of the diagonals to be 78 feet. Explain how you can use this measurement to tell whether the foundation will be rectangular.

  31. Solution: The diagonal divides the foundation into two triangles. Compare the square of the length of the longest side with the sum of the squares of the shorter sides of one of these triangles. Because 302 + 722 = 782, you can conclude that both the triangles are right triangles. The foundation is a parallelogram with two right angles, which implies that it is rectangular Building a foundation

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