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Triangle Properties Distance Congruent Similar Pythagorean Trigonometry Transit – measure Use angles and distance Similar Triangles Congruent Triangles Right Triangles and the Pythagorean Theorem Right Triangles and the Pythagorean Theorem Escher used right angles to form his artwork known as Waterfall. Contains three uses of the impossible tribar Created by British mathematician Roger Penrose (b 1931) in 954. Pythagorean Theorem 2 a + 2 b = 2 c Pythagoras of Samos Greek mathematician about 569 BC about 475 BC Pythagorean Theorem In any right angled triangle: • The area of the square whose side is the hypotenuse • = • The sum of the areas of the squares whose sides are the two legs The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation a2 + b2 = c2 Given a right triangle Pythagorean Theorem uses relationship of side lengths 1.c is the longest side of the triangle, called the "hypotenuse" 2.a and b are the legs Pythagorean Theorem Pythagorean Theorem 2 a + 2 b = 2 c President Garfield’s Proof 1881 One uses subtraction, the other addition. How can opposite operations prove the same thing? 2 a + 2 b 2 =c Algebraic Proof Area of Large Square A = (a+b) (a+b) Area of Pieces A=c*c A = 4 (1/2) ab Total Area of Pieces A = c2 + 2ab Both areas are equal Therefore, (a+b) (a+b) = c2 + 2ab a2+ + 2ab + b2 = a2+b2 = c2 Similarity of the triangles leads to the equality of ratios of corresponding sides: The first result equates the cosine of each angle θ and the second result equates the sines. These ratios can be written as: BC2 = AB x BH and AC2 = AB x AH BC2 + AC2 = ABxBH + ABxAH = AB (AH + BH) = AB2 The Pythagorean Theorem BC2 + AC2 = AB2 Pythagorean Theorem Given Points R (-4,5) and S (2, -1) Determine the length of Line RS. How high up the wall does the ladder reach? Broken Pole Napoleon’s Officer Given ∠DEG and ∠DEF are right angles ∠EDG ∠EDF Prove EF EG Solve for x. Solve for x. 13’ 12’ 7’ x’ Solve for x’. Pythagorean and a bit more • A diver hooked to a 25meter line is searching for the remains of a Spanish galleon in the Caribbean Sea. The sea is 20 meters deep and the bottom is flat. • What is the area of the circular region that the diver can explore? Pythagorean and a bit more • One of the sketches below shows the greatest area that you can enclose in a right angled corner with a rope of length s. • Which one? Explain your reasoning. Rationalize the Denominator • When you move a root out from the denominator of a fraction to the numerator • Equations are often not considered 'completely simplified' if there is still a radical in the denominator • The denominator should not have an irrational number • Fix by making the denominator rational • Called "Rationalizing the Denominator" 45-45 Triangle 45 a2 + b2 = c2 x2 + x 2 = c 2 2x2 = c2 IF x = 1 2 = c2 X 2 c 45 90 X Hippasus of Metapontum 30-60 triangle Pythagorean Triple •Set of 3 positive whole numbers •a, b, c – side lengths of a right triangle •a, b, c – satisfy the equation from the Pythagorean Theorem a2 + b2 = c2 Table of the first few primitive Pythagorean triples: r s a b c 2 1 3 4 5 3 2 5 12 13 4 1 15 8 17 4 3 7 24 25 5 2 21 20 29 5 4 9 40 41 6 1 35 12 37 6 5 11 60 61 7 2 45 28 53 Pythagorean triple scatterplot Plot of triples generated by Euclid's formula Pythagorean Inequalities The following triples represent the side lengths of a triangle. Determine whether the triangle is right, acute or obtuse. Obtuse 1. 5, 9, 12 Obtuse 2. 7, 24, 26 Acute 3. 13, 15, 17 Right 4. 3, 4, 5 Acute 5. 25, 25, 30 1. You've just picked up a ground ball at first base, and you see the other team's player running towards third base. How far do you have to throw the ball to get it from first base to third base, and throw the runner out? 2. You're locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. Your neighbor has a ladder that is 14’ long. Will you borrow the ladder? a c b c1 b1 a1 THE GENERAL EXTENSION TO PYTHAGORAS' THEOREM: If any 3 similar shapes are drawn on the sides of a right triangle, then the area of the shape on the hypotenuse equals the sum of the areas on the other two sides. But why? Pythagorean Work in Music • Music of the Spheres • Harmony through the ratio of string lengths Solve for the Unknown Pythagorean Theorem Spiral of Theodorus Spiral of Theodorus to 3 windings Geometry in Art Spherical Geometry Hyperbolic Geometry Parallax Parallax parallax angle = half of total angle L= half of total baseline -------------------------------------tan [ half of total angle] Similar Triangles Transit of Venus It takes about 7 hours for Venus to cross the Sun. Photos taken during the transit show Venus following its orbit around the Sun. To transit means to pass across or through somewhere. The tiny shadow of Venus on the Sun transits across the sun's disc. Why are Transits of Venus so rare? Simulation of the parallax of a star which is at a distance of 1 pc (closer to our Sun than any real star) Subtle change January to July