1. Keystone Geometry 1 The Pythagorean Theorem

2. Used to solve for the missing piece of a right triangle. Only works for a right triangle. Given any right triangle, A 2 + B 2 = C 2, where A and B are the legs of the triangle and C is the hypotenuse. The Pythagorean Theorem 2 A B C

3. Example 3 A B C In the following figure if A = 3 and B = 4, Find C. A 2 + B 2 = C 2 3 2 + 4 2 = C 2 9 + 16 = C 2 5 = C

4. Verifying the Pythagorean Theorem 4 Given a piece of graph paper, make a right triangle. Then make squares of the right triangle. Then find the square’s areas.

5. Pythagorean Theorem : Examples for finding the hypotenuse. 5 A=8, B= 15, Find C A=7, B= 24, Find C A=9, B= 40, Find C A=10, B=24, Find C A =6, B=8, Find C A B C C = 17 C = 25 C = 41 C = 26 C = 10

6. Finding the legs of a right triangle: 6 A B C In the following figure if B = 5 and C = 13, Find A. A 2 + B 2 = C 2 A 2 +5 2 = 13 2 A 2 + 25 = 169 A 2 = 144 A = 12

7. More Examples: 7 1) A=8, C =10, Find B 2) A=15, C=17, Find B 3) B =10, C=26, Find A 4) A=15, B=20, Find C 5) A =12, C=16, Find B 6) B =5, C=10, Find A 7) A =6, B =8, Find C 8) A=11, C=21, Find B A B C B = 6 B = 8 A = 24 C = 25 A = 8.7 C = 10

8. Given the lengths of three sides, how do you know if you have a right triangle? 8 A B C Given A = 6, B=8, and C=10, describe the triangle. A 2 + B 2 = C 2 6 2 +8 2 = 10 2 36 + 64 = 100 * This is true, so you have a right triangle.

9. Pythagorean Triples 9 Some right-angled triangles where all three sides are whole numbers called Pythagorean Triangles. The three whole number side-lengths are called a Pythagorean triple. The 3-4-5 triangle An example is a = 3, b = 4 and h = 5, called "the 3-4-5 triangle". We can check it as follows:

10. Pythagorean Triples  Not only is 3-4-5 a Pythagorean triple, but so is any multiple of 3-4-5.  3-4-5  6-8-10  12-16-20  15-20-25  18-24-30 10 Can you think of any others?? 5,12,137,24,2520,21,29

11. What happens if it the Pythagorean Theorem does NOT work?  If you do not have a picture nor an angle that you know for a fact is 90 degrees, then it is possible to have an acute or an obtuse triangle.  If A 2 + B 2 > C 2, you have an acute triangle.  If A 2 + B 2 < C 2, you have an obtuse triangle. 11

12. If A 2 + B 2 > C 2, it is an acute triangle. 12  Given A = 4, B = 5, and C = 6, describe the triangle. A 2 + B 2 = C 2 4 2 + 5 2 = 6 2 16 + 25 = 36 41 > 36, so we have an acute triangle. A B C

13. If A 2 + B 2 < C 2, it is an obtuse triangle. 13  Given A = 4, B = 6, and C =8, describe the triangle. A 2 + B 2 = C 2 4 2 + 6 2 = 8 2 16 + 36 = 64 52 < 64, so we have an obtuse triangle. A C B

14. Describe the following triangles as acute, right, or obtuse 14 1) A=9, B=40, C=41 2) A=10, B=15, C=20 3) A=2, B=5, C=6 4) A=12, B=16, C=20 5) A=11, B=12, C=14 6) A=2, B=3, C=4 7) A=1, B=7, C=7 8) A=90, B=120, C=150 A B C right acute obtuse right obtuse acute