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 don’t 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 society’s 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 θ 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 θ, 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.

Another Proof…

Proof … Let in ΔABC, 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 ΔACE 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

a² = c² - b²,a² + b² = c².

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

"abc”triangle 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 =½ab So all four of them combined is A = 4(½ab) = 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 Geologist’s 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

4miles

Ans 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, <F =<C = 90 degrees.

And hence we have proved that triangle is right angled at C .

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