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:
Pythagorean Theorem 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 .
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?
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
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