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Transcript
Aim: What is the Pythagorean Theorem & how do we use it? Do Now: Approximate 32 to the nearest tenth. Aim: Pythagorean Theorem Course: Applied Geometry 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 Radical sign 25 5 Square Root Radicand • 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. Aim: Pythagorean Theorem Course: Applied Geometry Yasoo, myfree Since I’ve a lot of time on It’s sohad important, they mynamed hands, Iitthought I’d look name after is me: Theat some properties of Theorem. a right triangle. Pythagoras. Pythagorean Hmm. . .In a right triangle . . . Cool, huh! B c a . . 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. c2 = a2 + b2 C b A 52 = 32 + 42 25 = 9 + 16 25 = 25 5 4 3 Aim: Pythagorean Theorem Course: Applied Geometry The Square of the What? F Pythagorean Theorem c2 = a2 + b2 A cc2 2 b2 2 b c a C B a2 2 Aim: Pythagorean Theorem Course: Applied Geometry 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. Pythagorean Theorem c2 = a2 + b2 852 = 842 + 132 7225 = 7056 + 169 7225 = 7225 c2 = a2 + b2 Aim: Pythagorean Theorem Course: Applied Geometry Model Problem Find the value of x. Round to nearest tenth. 8 20 x Pythagorean Theorem c2 = a2 + b2 202 = 82 + x2 400 = 64 + x2 336 = x2 x = 336 x = 18.3 Aim: Pythagorean Theorem Course: Applied Geometry Model Problem Use the triangle below to find the missing length. Round to nearest tenth. a c b Pythagorean Theorem c2 = a2 + b2 a = 3, b = 7, c = ?7.6 a = ?, 19.3, b = 23, c = 30 a = 1.2, b = ?3.3,, c = 3.5 Aim: Pythagorean Theorem Course: Applied Geometry Model Problem The hypotenuse of a right triangle is 25. If one leg is 20, the other leg is 1) 5 3) 15 2) 1025 4) 45 Which of the following could be the lengths of the sides of a right triangle? 1) 3,5,8 3) 2,4,6 2) 5,12,13 4) 5,5,5 Aim: Pythagorean Theorem Course: Applied Geometry 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 c = length of ladder = ? 13 b = distance from wall = 5’ a = height above ground = 12’ 12’ ? 5’ c2 = 122 + 52 c2 = 144 + 25 c2 = 169 c = 13 Cool! Aim: Pythagorean Theorem Course: Applied Geometry 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 350 m. a c2 = 62,500 + 122,500 c c2 = 185,000 c= dock 250 m. = b Aim: Pythagorean Theorem 185,000 c = 430.11626 c = 430 m. Course: Applied Geometry to nearest meter Model Problem A pole BD, 28 feet high, is perpendicular to the A pole ground. Two wires, BC and BA,each 35 feet long, are attached to the top of the pole and to stakes A C on the ground. If points A, D, and C are collinear. how far are the stakes A and C from each other? B c2 = a2 + b2 352 = 282 + DC2 28’ 2 1225 = 784 + DC 35’ 35’ A D ? C 441 = DC2 21 = DC AC = 2DC = 2(21) = 42’ Aim: Pythagorean Theorem Course: Applied Geometry Model Problem Find the value of x. 4 5 x = 17.89 ?8 4 16 Pythagorean Theorem c2 = a2 + b2 4 5 2 4 AB 2 2 x 2 82 162 80 = 16 + AB2 x2 = 320 64 = AB2 x = 17.89 x=8 Aim: Pythagorean Theorem Course: Applied Geometry Pythagorean Triplets Goo Goo Goo a b 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? Aim: Pythagorean Theorem Course: Applied Geometry Pythagorean Triplets Goo Goo a Goo b 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 Aim: Pythagorean Theorem Course: Applied Geometry