1. Geometry 9.3 Converse of the Pythagorean Theorem

2. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem2 Goals I can determine if a triangle is a right triangle. I can use the Pythagorean inequalities to determine if a triangle is acute or obtuse.

3. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem3 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. If  ABC is a right triangle, then a 2 + b 2 = c 2 a b c A B C

4. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem4 Converse of Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If a 2 + b 2 = c 2, then  ABC is a right triangle. a b c A B C

5. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem5 Example 1 Is  POD a right triangle? P O D 30 16 34 Yes! Longest Side

6. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem6 Reminder

7. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem7 Example 2 Is  HUG a right triangle? H U G 5 10 Which segment is the longest? HG Yes!

8. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem8 Example 3 Is  SAD a right triangle? S A D 9 12 Which segment is the longest? SD No! 20

9. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem9 Your Turn. Is  RST a right  ? R S T 26 10 24 Yes it is.

10. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem10 Triangle Inequality Theorem In a triangle, the sum of any two sides is greater than the third side. 4 5 7 4 + 7 > 5 4 + 5 > 7 5 + 7 > 4

11. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem11 Triangle Inequality Theorem 5 10 4 This is not a triangle since 5 + 4 < 10.

12. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem12

13. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem13 a b c Begin with a right triangle… a 2 + b 2 = c 2

14. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem14 a b c a c a and b have not changed. a 2 + b 2 has not changed. c got smaller. c 2 got smaller. and… The right angle gets smaller: it is acute. Rotate side a in. c 2 = a 2 + b 2 c 2 < a 2 + b 2

15. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem15 Theorem 9.6 If the square of the length of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. A B C a b c c 2 < a 2 + b 2

16. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem16 a b c Take another right triangle… a 2 + b 2 = c 2

17. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem17 a b c a c a and b have not changed. a 2 + b 2 has not changed. c got larger. c 2 got larger. and… The right angle gets larger: it is obtuse. Rotate side a out. c 2 = a 2 + b 2 c 2 > a 2 + b 2

18. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem18 Theorem 9.6 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. c 2 > a 2 + b 2 A B C a b c

19. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem19 Example 4 The sides of a triangle measure 5, 7, and 11. Classify it as acute, right, or obtuse. Solution: The longest side is 11. 11 2 ? 5 2 + 7 2 121 ? 25 + 49 121 > 74 Obtuse

20. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem20 Example 5 The sides of a triangle are 17, 20, and 25. Classify the triangle. Solution: 25 2 ? 17 2 + 20 2 625 ? 689 625 < 689 Acute

21. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem21 Example 6 Classify this triangle. ? ? Right 

22. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem22 Example 7 Classify this triangle. It isn’t a triangle! 6 +8 < 16. 6 8 16

23. September 23, 2015Geometry 9.3 Converse of the Pythagorean Theorem23 Summary If c 2 = a 2 + b 2, RIGHT . If c 2 < a 2 + b 2, ACUTE . If c 2 > a 2 + b 2, OBTUSE . The last two can be very confusing; don’t get them mixed up.