1. 9.3 Converse of the Pythagorean Theorem
Geometry
9.3 Converse of the Pythagorean Theorem

2. Geometry 9.3 Converse of the Pythagorean Theorem
Goals
Determine if a triangle is a right triangle.
Use the Pythagorean inequalities to determine if a triangle is acute or obtuse.
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

3. Geometry 9.3 Converse of the 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 a2 + b2 = c2 a b c A B C
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

4. 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 a2 + b2 = c2, then ABC is a right triangle. a b c A B C
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

5. Geometry 9.3 Converse of the Pythagorean Theorem
Example 1
Is POD a right triangle? P O D 30 16 34
Longest Side
Yes!
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

6. Geometry 9.3 Converse of the Pythagorean Theorem
Reminder
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

7. Geometry 9.3 Converse of the Pythagorean Theorem
Example 2
Which segment is the longest?
Is HUG a right triangle? HG H U G 5 10
Yes!
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

8. Geometry 9.3 Converse of the Pythagorean Theorem
Example 3
Which segment is the longest?
Is SAD a right triangle? SD S A D 9 12 20
No!
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

9. Geometry 9.3 Converse of the Pythagorean Theorem
Your Turn.
Is RST a right ? S 24 10
Yes it is. R T 26
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

10. Triangle Inequality Theorem
In a triangle, the sum of any two sides is greater than the third side.
4 + 7 > 5
4 + 5 > 7
5 + 7 > 4 7 4 5
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

11. Triangle Inequality Theorem
This is not a triangle since < 10. 4 5 10
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

12. Geometry 9.3 Converse of the Pythagorean Theorem
Inequalities
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

13. Begin with a right triangle…
a2 + b2 = c2 c a b
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

14. Geometry 9.3 Converse of the Pythagorean Theorem
Rotate side a in.
a and b have not changed.
a2 + b2 has not changed.
c got smaller.
c2 got smaller.
and…
The right angle gets smaller: it is acute. c c a a b
c2 < a2 + b2
c2 = a2 + b2
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

15. Geometry 9.3 Converse of the Pythagorean Theorem
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
c2 < a2 + b2
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

16. Take another right triangle…
a2 + b2 = c2 c a b
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

17. Geometry 9.3 Converse of the Pythagorean Theorem
Rotate side a out.
a and b have not changed.
a2 + b2 has not changed.
c got larger.
c2 got larger.
and…
The right angle gets larger: it is obtuse. c a c a b
c2 > a2 + b2
c2 = a2 + b2
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

18. Geometry 9.3 Converse of the Pythagorean Theorem
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. A B C a b c
c2 > a2 + b2
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

19. Geometry 9.3 Converse of the Pythagorean Theorem
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.
112 ?
121 ?
121 > 74
Obtuse
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

20. Geometry 9.3 Converse of the Pythagorean Theorem
Example 5
The sides of a triangle are 17, 20, and 25. Classify the triangle.
Solution:
252 ?
625 ? 689
625 < 689
Acute
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

21. Geometry 9.3 Converse of the Pythagorean Theorem
Example 6
Classify this triangle. ? ?
Right 
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

22. Geometry 9.3 Converse of the Pythagorean Theorem
Example 7
Classify this triangle.
It isn’t a triangle! 6 +8 < 16. 16 6 8
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem

23. Geometry 9.3 Converse of the Pythagorean Theorem
Summary
If c2 = a2 + b2, RIGHT .
If c2 < a2 + b2, ACUTE .
If c2 > a2 + b2, OBTUSE .
The last two can be very confusing; don’t get them mixed up.
February 19, 2019
Geometry 9.3 Converse of the Pythagorean Theorem