1. 8.1 Pythagorean Theorem and Its Converse SOL: G8 Objectives: The Student Will… Use the Pythagorean Theorem to determine the measure of missing legs and hypotenuses. Use the converse of the Pythagorean Theorem, to determine whether the measures will form a right triangle.

2. Pythagorean Theorem If you have a right triangle, then Leg 2 + Leg 2 = Hypotenuse 2 Hypotenuse Leg If the triangle is a right triangle, then a 2 + b 2 = c 2 A B C a c b

3. Example 1: Find x. 20 2 + 37.5 2 = x 2 13 2 + 23 2 = x 2 400 + 1406.25 = x 2 1806.25 = x 2 42.5 ≈ x 169 + 529 = x 2 698 = x 2 26.41968963 = x26.4 ≈ 37.5 20 x

4. Example 2: Find x and d. 6 2 + x 2 = 10 2 36 + x 2 = 100 -36 - 36 x 2 = 64 x = 8 3 2 + d 2 = 6 2 9 + d 2 = 36 -9 - 9 d 2 = 27 d ≈ 5.196152423 d ≈ 5.2

5. Study Guide pg 357 Find x. 1.)2.) 3.)4.)

6. Study Guide pg 357 Find x. 5.)6.)

7. Example 3: 2 2 + 2 2 = x 2 4 + 4 = x 2 8 = x 2 2.828427125 ≈ x Carson City, Nevada, is located at about 120 degrees longitude and 39 degrees latitude. Use the lines of longitude and latitude to find the degree distance to the nearest tenth degree if you were to travel directly from NASA Ames to Carson City, Nevada. Carson City, Nevada 2 2 8 = x 2 2.8 degrees ≈ x

8. Converse of the Pythagorean Theorem If Leg 2 + Leg 2 = Hypotenuse 2, then you have a right triangle If a 2 + b 2 = c 2, then the triangle is a right triangle.

9. Example 4: Are the following Right Triangles 17 8 15 8 2 + 15 2 = 17 2 64 + 225 = 289 ? ? 289 = 289 31 20 21 20 2 + 21 2 = 31 2 400 + 441 = 961 ? ? 841 = 961 Yes, This is a right triangle No, Not a right triangle

10. Example 5: Verify that ΔABC is a right Triangle Recall the distance formula: AB = (-9 – 1) 2 + (-3 - -1) 2 d = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = (-10) 2 + (-2) 2 = 100 + 4= 104 AC = (-9 – -3) 2 + (-3 - -7) 2 = (-6) 2 + (4) 2 = 36 + 16 = 52 BC = (1 – -3) 2 + (-1 - -7) 2 = (4) 2 + (6) 2 = 16 + 36 = 52 AC 2 + BC 2 = AB 2 ? 104 = 104 52 ( ) 2 + 52 ( ) 2 = 104 ( ) 2 52 + 52 = 104 Yes, this is a right triangle

11. Example 6: Determine whether each set of measures are the sides of a right triangle. Then state whether they form a Pythagorean triple. Right Triangle, Pythagorean Triple Not whole numbers, Not Pythagorean Triple Not a Right Triangle, Not Pythagorean Triple a.) 9, 12, and 15b.) 21, 42, and 54 225 = 225 9 2 + 12 2 = 15 2 ? 21 2 + 42 2 = 54 2 ? 2205 = 2916 c.) 4 3, 4, and 8 (4 3 ) 2 + 4 2 = 8 2 64 = 64

12. Pythagorean Triple Is three whole numbers that satisfy the equation a 2 + b 2 = c 2, where c is the greatest number. 7 2 + 24 2 = 25 2 Pythagorean Triple Example: 7, 24, 25 625 = 625 ____________ ___ _______________

13. Study Guide pg 358 Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean Triple. 1.) 30, 40, 502.) 20, 30, 40 3.) 18, 24, 304.) 6, 8, 9 5.) 6.) 10, 15, 20

14. Study Guide pg 358 Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean Triple. 7.) 8.) 9.) 9, 40, 41

15. Study Guide pg 358 A family of Pythagorean Triples consists of multiples of known triples. For each Pythagorean Triple, find two triples in the same family. 10.) 3, 4, 511.) 5, 12, 13 12.) 7, 24, 25