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  • Pythagoras

  • Hello and welcome to the adventure of discovering the Pythagorean Theorem. The Pythagorean Theorem is one of the greatest theorems known today. This discovery was credited to Pythagoras of Samos. Pythagoras believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured. So as you embark on your discovery remember that great minds think alike and dont give up no matter what the cost.

  • History of Pythagoras

    Pythagoras founded a philosophical and religious school/society in Croton (now spelled Crotone, in southern Italy)

    His followers were commonly referred to as Pythagoreans.

    The members of the inner circle of the society were called the mathematikoi

    The members of the society followed a strict code which held them to being vegetarians and have no personal possessions.

  • History of Pythagoras (cont.)

    There is not much evidence of Pythagoras and his societys work because they were so secretive and kept no records .

    One major belief was that all things in nature and all relations could be reduced to number relations .

  • Pythagoras and Music Pythagoras made important developments

    in music and astronomy Observing that plucked strings of

    different lengths gave off different tones, he came up with the musical scale still used today.

  • Pythagoras and Math Pythagoras made many contributions

    to the world of math including: Studies with even/odd numbers Studies involving Perfect and Prime

    Numbers Irrational Numbers Various theorems/ideas about triangles,

    parallel lines, circles, etc. Of course THE PYTHAGOREAN THEOREM

  • Proof by similar triangles. This proof is based on the proportionality of the

    sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangleABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as in the figure. By a similar reasoning,thetriangle CBH is also similar to ABC. The proof of similarity of the triangles requires the

    Triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to theparallel postulate.

    Similarity of the triangles leads to the equality of ratios of corresponding sides:

  • The first result equatesthe cosines of the angles , whereas the

    second result equates their sines. These ratios can be written as

    Summing these two equalities results in

    which, after simplification, expresses the Pythagorean theorem:

    https://en.wikipedia.org/wiki/Cosine

  • Pythagorean Theorem proof.

  • Another Proof

  • Proof Let in ABC, angle C = 90. As usual, AB = c, AC = b, BC =

    a. Define points D and E on AB so that AD = AE = b. By construction, C lies on the circle with center A and radius

    b. Angle DCE subtends its diameter and thus is right: DCE = 90. It follows that BCD = ACE. Since ACE is isosceles, CEA = ACE.

    Triangles DBC and EBC share DBC. In addition, BCD = BEC. Therefore, triangles DBC and EBC are similar. We have BC/BE = BD/BC, or

    a / (c + b) = (c - b) / a. And finally

    a = c - b,a + b = c.

  • Proof of the theorem There are many other way to prove the

    theorem but we will refer to this as our proof. Take a look at this diagram ... it has that

    "abctriangle in it (four of them actually): Now let's add up the areas of all the smaller

    pieces: First, the smaller (tilted) square has an area

    of A = c2 And there are four triangles, each

    one has an area of A =ab So all four of them combined is A = 4(ab) = 2ab So, adding up the tilted

    square and the 4 triangles gives: A = c2+2ab

  • The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:(a+b)(a+b) = c2+2abNOW, let us rearrange this to see if we can get the pythagoras theorem:Start with: (a+b)(a+b)=c2 + 2ab Expand (a+b)(a+b): a2 + 2ab + b2=c2 + 2ab

    Subtract "2ab" from both sides: a2 + b2 = c2

  • Discovering the distance from which ladder is to be kept .

    The tallest building in Bangalore is 356 feet. The average height for a fire engine is 12 feet. If the fire department is required to be 50 feet away from the building how long (in feet) must the ladder be to reach someone on the roof of the tallest building? Round to two decimal places.

    The distance of the ladder is FIXED. The distance of the burning floor is FIXED. (from the ground.) By using the theorem we could change the distance between the foot of the ladder and the foot of the building .

    25 feet

    24 feet

    Daily Life Applications

    Ans. 7 feet

  • Finding the quickest way(A Geologists best friend)

    A geologist is looking for gold. To reach his destination he must traverse around a swamp. He heads south 3 miles and then heads east 4 miles. If the geologist could cross the swamp how much distance (in miles) would he have saved?

    3 miles

    4milesAns 2 miles..

  • Converse of the Pythagoras theorem In order to prove the converse of the

    Pythagorean Theorem, we need to prove that in the figure is a right angle. Now, we discuss the proof.

    Theorem In a triangle with sides , and (see

    figure above), if a2 + b2 = c2 holds, then is a right triangle with a right angle

    at C . Proof Let DEF be a triangle such that EF=a ,

    DF=b and right angled at F . If we let DE= x , since DEF is a right triangle, by the Pythagorean Theorem

    a2 + b2 = x2 ..(1). But from the supposition, a2 + b2 = c2 (2).

  • From (1) and (2) x2 =c2 Since and are both positive ,we

    can therefore conclude that x =c. This means that length of the

    three corresponding pairs of sides of triangle ABC and triangle DEF are equal.

    Therefore, by SSS Congruence,

    ABC DEF Since and are corresponding

    angles,

  • http://www.cut-the-knot.org/pythagoras/index.shtml (There are more than 144 proofs on this site.)

    Thank You.Prepared by : Raneet P Sahoo.

    Class X Roll no. 10