In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle ( right-angled triangle ). In terms of areas, it states: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

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The theorem can be written as an equation relating the lengths of the sides a , b and c , often called the Pythagorean equation : where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

Right-angled Triangle:

Right-angled Triangle

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History

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The Pythagorean theorem is named after the Greek mathematician Pythagoras(569 B.C.?-500 B.C.?), who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they fitted it into a mathematical framework.

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The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n -dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

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Biography

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Pythagoras

Biography:

Biography Pythagoras was born in 572BC on the island of Samos, Greece. In about 530BC Pythagoras left Samos in hatred for its ruler Polycrates and settled in Cretona, Italy. He joined a religious group known as the Pythagoreans. He formed a philosophical and religious school where they studied mathematics, science and music. This attracted many followers. When involved with this group he discovered what is now known as Pythagoras’ Theorem. Also during his time there he out the mathematics of octaves and harmony. Because of the secrecy in the group there is nothing of Pythagoras’ writings or books. Pythagoras was murdered at the age of 77, in 495BC and the religious school was separated.

Contemporary importance:

Contemporary importance Pythagoras’ theorem has been used over thousands of years for many different aspects of human life. Today his theorem is used mainly in building, architecture, carpenter, navigation, astronomy and many other fields of work that involve mathematical calculations. Each of these fields uses his theorem to try and decide either the hypotenuse or the two other sides in a right angled triangle. -Builders use Pythagoras’ theorem to work out dimensions of different aspects of their constructions. This allows them to work out the exact requirements of building materials needed. -Architects use his theorem to work out designs for the builders to use. His theory may be used to work out exact lengths of roofs, also the framework of a house just to name a few. -Carpenter’s uses his theorem to interprete the size of sides of their timber structures eg: corner furniture. -Navigators and astronomers use his theorem to establish distances between planets, towns, countries and stars.

Relevance:

Relevance Pythagoras is often described as one of the pure mathematicians of his time and an extremely important figure in the expansion of mathematics. Pythagoras’ theorem is studied from Year 8 to Year 12 in NSW schools. Students today often wonder why geometry is so important. It allows people to think more logically and as I have shown in Contemporary Importance his theorem is used in numerous jobs and work areas. Students pursuing technical majors in college are expected to understand and extend this knowledge on geometry. Geometry proofs are also an important way to increase disciplined . As you can see geometry is still just as important now as it was in Pythagoras’ time.

Mathematical achievements:

Mathematical achievements Pythagoras has contributed various theories to geometry, algebra, number .etc. All of these theories were discovered during Pythagoras’ time with the Pythagoreans. Pythagoras’ theory is : The square on the hypotenuse in any right-angled triangle is equal to the sum of the squares on the other two sides. So for example: A B C a b c 4 3 c Using the formula a ²+b²=c² find side “c” on the triangle DEF D E F 4 ² + 3² = c² 16 + 9 = c² 25 = c² √25 = c² c = 5

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Pythagoras believed: All things are numbers. Mathematics is the basis for everything, and geometry is the highest form of mathematical studies. The physical world can understood through mathematics. Certain symbols have a mystical significance. All members of the society should observe strict loyalty and secrecy.

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4. The world depends upon the interaction of opposites, such as male and female, lightness and darkness, warm and cold, dry and moist, light and heavy, fast and slow. 5. The soul resides in the brain, and is immortal. It moves from one being to another, sometimes from a human into an animal, through a series of reincarnations called transmigration until it becomes pure. Pythagoras believed that both mathematics and music could purify.

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Some of the students of Pythagoras eventually wrote down the theories, teachings and discoveries of the group, but the Pythagoreans always gave credit to Pythagoras as the Master for: The five regular solids (tetrahedron, cube, octahedron, icosahedrons, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two.

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2. The sum of the angles of a triangle is equal to two right angles. 3. Pythagoras taught that Earth was a sphere in the center of the Kosmos (Universe), that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus .

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4. Pythagoras studied odd and even numbers, triangular numbers, and perfect numbers. Pythagoreans contributed to our understanding of angles, triangles, areas, proportion, polygons, and polyhedra . 5. Pythagoras also related music to mathematics. He had long played the seven string lyre, and learned how harmonious the vibrating strings sounded when the lengths of the strings were proportional to whole numbers, such as 2:1, 3:2, 4:3. Pythagoreans also realized that this knowledge could be applied to other musical instruments.

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The reports of Pythagoras' death are varied. He is said to have been killed by an angry mob, to have been caught up in a war between the Agrigentum and the Syracusans and killed by the Syracusans, or been burned out of his school in Crotona and then went to Metapontum where he starved himself to death. At least two of the stories include a scene where Pythagoras refuses to trample a crop of bean plants in order to escape, and because of this, he is caught. The Pythagorean Theorem is a cornerstone of mathematics, and continues to be so interesting to mathematicians that there are more than 400 different proofs of the theorem, including an original proof by President Garfield.

Statement of the Theorem:

Statement of the Theorem

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It is believed that the statement of Pythagorean's Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. The Pythagorean Theorem relates to the three sides of a right triangle. It states that c2=a2+b2, C is the side that is opposite the right angle which is referred to as the hypoteneuse. a and b are the sides that are adjacent to the right angle. In essence, the theorem simply stated is: the sum of the areas of two small squares equals the area of the large one.

The theorem states that: :

The theorem states that: For any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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Verification of Theorem

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1. cut a triangle with base 4 cm and height 3 cm 0 1 2 3 4 5 4 cm 0 1 2 3 4 5 3 cm 2. measure the length of the hypotenuse 0 1 2 3 4 5 Now take out a square paper and a ruler. 5 cm

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Consider a square PQRS with sides a + b a a a a b b b b c c c c Now, the square is cut into - 4 congruent right-angled triangles and - 1 smaller square with sides c Proof of Pythagoras’ Theorem P Q R S

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a + b a + b A B C D Area of square ABCD = ( a + b ) 2 b b a b b a a a c c c c P Q R S Area of square PQRS = 4 + c 2 a 2 + 2ab + b 2 = 2ab + c 2 a 2 + b 2 = c 2

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Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." The Pythagorean Theorem asserts that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a 2 + b 2 = c 2 The figure above at the right is a visual display of the theorem's conclusion. The figure at the left contains a proof of the theorem, because the area of the big, outer, green square is equal to the sum of the areas of the four red triangles and the little, inner white square: c 2 = 4(ab/2) + (a - b) 2 = 2ab + (a 2 - 2ab + b 2 ) = a 2 + b 2

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Animated Proof of the Pythagorean Theorem Below is an animated proof of the Pythagorean Theorem. Starting with a right triangle and squares on each side, the middle size square is cut into congruent quadrilaterals (the cuts through the center and parallel to the sides of the biggest square). Then the quadrilaterals are hinged and rotated and shifted to the big square. Finally the smallest square is translated to cover the remaining middle part of the biggest square. A perfect fit! Thus the sum of the squares on the smaller two sides equals the square on the biggest side. Afterward, the small square is translated back and the four quadrilaterals are directly translated back to their original position. The process is repeated forever.

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Pythagorean Triplets

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The sides of a right triangle follows the Pythagorean Theorem, a2 + b2 = c2 where a and b are the lengths of the legs of the right triangle while c is the length of the hypothenuse. A right triangle with sides of lengths 3, 4 and 5 is a special right triangle in that all the sides have whole number lengths. The three numbers 3, 4 and 5 forms a Pythagorean triplet or Pythagorean triple.

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A Pythagorean triplet is a set of three whole numbers where the sum of the squares of the first two is equal to the square of the third number. Below are examples of Pythagorean triplets: 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61

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One equation satisfying a Pythagorean Triplet A, B, C is Given A is odd, then B = (A2 - 1)/2 C = (A2 + 1)/2 Another equation derived by Plato was (m2+1)2 = (m2-1)2 + (2m)2 where m is a natural number. The above equation is called Plato's Formula.

Why Memorize Pythagorean Triples?:

Why Memorize Pythagorean Triples? Remember how much time it took to figure out 8 x 8 before you memorized it? (8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64) Think of all the work involved to solve this problem: a 2 + b 2 = c 2 3 2 + 4 2 = x 2 9 + 16 = x 2 3 4 x 25 = x 2 5 = x Wouldn’t it be nice to just know this is 5?

Good Pythagorean Triples to Memorize::

Good Pythagorean Triples to Memorize: And multiples of each, like: 3x2, 4x2, 5x2 6 8 10

Application Of Pythagoras Theorem:

Application Of Pythagoras Theorem

Applications:

Applications The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications. The theorem is invaluable when computing distances between two points, such as in navigation and land surveying. Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand.

Baseball Problem:

Baseball Problem A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

Baseball Problem:

Baseball Problem The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base?

Baseball Problem:

Baseball Problem To use the Pythagorean theorem to solve for x, find the right angle. Which side is the hypotenuse? Which sides are the legs? Now use: a 2 + b 2 = c 2

Baseball Problem Solution:

Baseball Problem Solution The hypotenuse is the distance from home to second, or side x in the picture. The legs are from home to first and from first to second. Solution: x 2 = 90 2 + 90 2 = 16,200 x = 127.28 ft

Ladder Problem:

Ladder Problem A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?

Ladder Problem Solution:

Ladder Problem Solution First draw a diagram that shows the sides of the right triangle. Label the sides: Ladder is 25 m Distance from house is 7 m Use a 2 + b 2 = c 2 to solve for the missing side. Distance from house: 7 meters

Ladder Problem Solution:

Ladder Problem Solution 7 2 + b 2 = 25 2 49 + b 2 = 625 b 2 = 576 b = 24 m How did you do?

Indirect Measurement:

Indirect Measurement Support Beam: The skyscrapers shown on page 535 are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.

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Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).

Solution::

(hypotenuse)2 = (leg)2 + (leg)2 x 2 = (23.26) 2 + (47.57) 2 x 2 = √ (23.26) 2 + (47.57) 2 x ≈ 13 Pythagorean Theorem Substitute values. Multiply and find the positive square root. Use a calculator to approximate. Solution:

Lets learn with Fun:

Lets learn with Fun

Pythagoras Board Game:

Pythagoras Board Game Rules: To begin, roll 2 dice. The person with the highest sum goes first. To move on the board, roll both dice. Substitute the numbers on the dice into the Pythagorean Theorem for the lengths of the legs to find the value of the length of the hypotenuse. Using the Pythagorean Theorem a²+b²=c², a player moves around the board a distance that is the integral part of c. For example, if a 1 and a 2 were rolled, 1²+2²=c²; 1+4=c²; 5=c²; Since c = √5 or approximately 2.236, the play moves two spaces. Always round the value down. When the player lands on a ‘?’ space, a question card is drawn. If the player answers the question correctly, he or she can roll one die and advance the resulting number of places. Each player must go around the board twice to complete the game. A play must answer a ‘?’ card correctly to complete the game and become a Pythagorean

Pythagoras Board Game:

Pythagoras Board Game What are the lengths of the legs of a 30-60-90 degree triangle with a hypotenuse of length 10? Answer: 5 and 5√3 If you hiked 3 km west and the 4 km north, how far are you from your starting point? Answer: 5 km The square of the ______ of a right triangle equals the sum of the squared of the lengths of the two legs. Answer: hypotenuse Find the missing member of the Pythagorean triple (7, __,, 25). Answer: 24 What is the length of the legs in a 45-45-90 degree right triangle with hypotenuse of length √2? Answer: 1 Using a²+b²=c, find b if c = 10 and a = 6 Answer: b=8² True or false? Pythagoras lives circa A.D. 500 Answer: false (500 B.C.) Have the person to your left pick two numbers for the legs of a right triangle. Compute the hypotenuse Can an isosceles triangle be a right triangle? Answer: yes Pythagoras was of what nationality? Answer: Greek Is (7, 8, 11) a Pythagorean triple? Answer: no How do you spell Pythagoras? The Pythagorean Theorem is applicable for what type of triangle? Answer: a right triangle What is the name of the school that Pythagoras founded? Answer: The Pythagorean School True or false? Pythagoras considered number to be the basis of creation? Answer: true True or false? Pythagoras formulated the only proof of the Pythagorean Theorem? Answer: false (there are about 400 possible proofs)

Thank You:

Thank You By- Rashmi Sharma VIII-A

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