Aim: What is the Pythagorean Theorem & how do we use it?

1. Aim: What is the Pythagorean Theorem & how do we use it? Do Now:

2. The Radical Square Root Radical sign Radicand Square Root The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product that is the original number. • Every positive number has a positive and negative sq. root. • The principal Sq. Root of a number is the positive sq. root. • A rational number can have a rational or irrational sq. rt. • An irrational number can only have an irrational root.

3. Since I’ve had a lot of free time on my hands, I thought I’d look at some properties of a right triangle. Hmm. . .In a right triangle . . . B C A Yasoo, my name is Pythagoras. It’s so important, they named it after me: The Pythagorean Theorem. Cool, huh! . . the square of the length of the hypotenuse c is equal to the sum of the squares of the lengths of the other two sides a and b. c a c2 = a2 + b2 b 52 = 32 + 42 5 25= 9+ 16 4 25= 25 3

4. Pythagorean Theorem c2 = a2 + b2 The Square of the What? c2 b2 a2

5. Pythagorean Theorem c2 = a2 + b2 Model Problem A right triangle has sides of lengths 20, 29, and 21. Which of these is the length of the hypotenuse? 29 Prove that a triangle with sides 13, 84 and 85 is a right triangle. 852 = 842 + 132 7225 = 7056 + 169 7225 = 7225 c2 = a2 + b2

6. Pythagorean Theorem c2 = a2 + b2 x = x = Model Problem Find the value of x. Round to nearest tenth. 20 8 x 202 = 82 + x2 400 = 64 + x2 336 = x2

7. Pythagorean Theorem c2 = a2 + b2 Model Problem Use the triangle below to find the missing length. Round to nearest tenth. c a b a = 3, b = 7, c = ? 7.6 a = ?, b = 23, c = 30 19.3, a = 1.2, b = ? , c = 3.5 3.3,

8. Model Problem The hypotenuse of a right triangle is 25. If one leg is 20, the other leg is Which of the following could be the lengths of the sides of a right triangle?

9. ? 12’ 5’ Cool! A ladder is placed 5 feet from the foot of a wall. The top of the ladder reaches a point 12 feet above the ground. Find the length of the ladder. c2 = a2 + b2 13 c = length of ladder = ? b = distance from wall = 5’ a = height above ground = 12’ c2 = 122 + 52 c2 = 144 + 25 c2 = 169 c = 13

10. c = A city park department rents paddle boats at docks near each entrance to the park. About how far to the nearest meter, is it to paddle from one dock to the other? c2 = a2 + b2 dock c2 = 3502 + 2502 c2 = 62,500+ 122,500 c 350 m. a c2 = 185,000 c = 430.11626 dock 250 m. = b c = 430 m. to nearest meter

11. B 28’ A D C ? Model Problem A pole c2 = a2 + b2 352 = 282 + DC2 1225 = 784 + DC2 35’ 35’ 441 = DC2 21 = DC AC = 2DC = 2(21) = 42’

12. ? Pythagorean Theorem c2 = a2 + b2 Model Problem Find the value of x. x = 17.89 8 4 16 80 = 16 + AB2 x2 = 320 64 = AB2 x = 17.89 x = 8

13. Goo Goo Goo Pythagorean Triplets ab c 3 4 5 5 12 13 8 15 17 For the Pythagorean Theorem, commonly used numbers that “work nicely” - and multiples of these Triplets There are others. Can you come up with one?

14. Goo Goo Goo Pythagorean Triplets ab c Find the 3rd side that would make the following pair a Pythagorean Triplet. 9, 41 and ? For the Pythagorean Theorem, commonly used numbers that “work nicely” - and multiples of these Triplets