2. Course: Applied Geometry Aim: Pythagorean Theorem Aim: What is the Pythagorean Theorem & how do we use it? Do Now:

3. Course: Applied Geometry Aim: Pythagorean Theorem 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. The Radical Square Root Radicand Radical sign 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.

4. Course: Applied Geometry Aim: Pythagorean Theorem Yasoo, my name is Pythagoras... 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. 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... 4 3 5 a b c CA B c 2 = a 2 + b2b2 5 2 = 3 2 + 4 2 25 = 9 + 16 25 = 25 It’s so important, they named it after me: The Pythagorean Theorem. Cool, huh!

5. Course: Applied Geometry Aim: Pythagorean Theorem The Square of the What? c2c2 b2b2 a2a2 c 2 = a 2 + b 2 Pythagorean Theorem

6. Course: Applied Geometry Aim: Pythagorean Theorem A right triangle has sides of lengths 20, 29, and 21. Which of these is the length of the hypotenuse? Prove that a triangle with sides 13, 84 and 85 is a right triangle. 85 2 = 84 2 + 13 2 7225 = 7056 + 169 7225 = 7225 29 Model Problem c 2 = a 2 + b 2 Pythagorean Theorem

7. Course: Applied Geometry Aim: Pythagorean Theorem Find the value of x. Round to nearest tenth. 20 8 x 20 2 = 8 2 + x 2 400 = 64 + x 2 336 = x 2 x = Model Problem c 2 = a 2 + b 2 Pythagorean Theorem

8. Course: Applied Geometry Aim: Pythagorean Theorem Use the triangle below to find the missing length. Round to nearest tenth. c a b Model Problem a = 3, b = 7, c = ? a = ?, b = 23, c = 30 c 2 = a 2 + b 2 Pythagorean Theorem a = 1.2, b = ?, c = 3.5 7.6 19.3, 3.3,

9. Course: Applied Geometry Aim: Pythagorean Theorem 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?

10. Course: Applied Geometry Aim: Pythagorean Theorem c = length of ladder = ? b = distance from wall = 5’ a = height above ground = 12’ 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. 5’ 12’ ? c 2 = a 2 + b 2 c 2 = 12 2 + 5 2 c 2 = 144 + 25 c 2 = 169 c = 13 Cool! 13

11. Course: Applied Geometry Aim: Pythagorean Theorem 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? 350 m. 250 m. c 2 = a 2 + b 2 c 2 = 350 2 + 250 2 c 2 = 62,500 + 122,500 c 2 = 185,000 c = c = 430.11626 c = 430 m. to nearest meter c a = b dock

12. Course: Applied Geometry Aim: Pythagorean Theorem c 2 = a 2 + b 2 35 2 = 28 2 + DC 2 1225 = 784 + DC 2 441 = DC 2 Model Problem A pole B D 28’ CA 35’ ? 21 = DC AC = 2DC = 2(21) = 42’

13. Course: Applied Geometry Aim: Pythagorean Theorem Model Problem Find the value of x. 416 x ? 80 = 16 + AB 2 64 = AB 2 x = 8 c 2 = a 2 + b 2 Pythagorean Theorem x = 17.89 x 2 = 320 8 = 17.89

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

15. Course: Applied Geometry Aim: Pythagorean Theorem Pythagorean Triplets Goo Triplets For the Pythagorean Theorem, commonly used numbers that “work nicely” - and multiples of these Triplets a b c Find the 3 rd side that would make the following pair a Pythagorean Triplet. 9, 41 and ?