coresub math att 4pythagoreantheorem

8/9/2019 CoreSub Math Att 4PythagoreanTheorem

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To day we are teaching about


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Pythagorean Theorem

Pythagoras (~580-500 B.C.)

He was a Greek philosopher responsible for important

developments in mathematics, astronomy and the theory of

music.


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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

3cm

2. measure the length of

the hypotenuse

0

1

2

3

4

5

Now take out a square paper and a ruler.

5cm


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Consider a square PQRS with sides a + b

a

a

a

a

b

b

b

bc

c

cc

Now , t he s quar e i s cu t i n t o- 4 con g ruen t r i gh t -ang led t r i ang les a n d- 1 sm a l le r squa re w i th s idesc

Proof of Pythagoras Theorem

P Q

R S


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a + b

a + b

A B

CD

Area of square ABCD

= (a + b) 2

b

b

a b

b

a

a

a

cc

cc

P Q

RS

Area of square PQRS

= 4 + c2ab

2

a 2 + 2ab + b 2 = 2ab + c 2

a2 + b2 = c2


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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:

a2 + b2 = c2

The figure above at the right is a visual display of the theorem'sconclusion. 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:

c2 = 4(ab/2) + (a - b)2 = 2ab + (a2 - 2ab + b2) = a2 + b2


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Animated Proof of the Pythagorean TheoremBelow 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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Animated Proof of the Pythagorean

Theorem


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Pythagorean Theorem

Over 2,500 years ago, a Greek mathematician named Pythagoras developed a proof thatthe relationship between the hypotenuse and the legs is true for allright triangles.

In any right triangle, the square of the length of the hypotenuse is equal to the sumof the squares of the lengths of the legs."

This relationship can be stated as:

and is known as the

Pythagorean Theorem

a, b are legs.c is the hypotenuse (across

from the right angle).

There are certain sets of numbers that have a very special property. Not only do

these numbers satisfy the Pythagorean Theorem, but any multiples of these numbersalso satisfy the Pythagorean TheoremFor example: the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you

multiply all three numbers by 2 (6, 8, and 10), these new numbers ALSO satisfy thePythagorean theorem.


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If we think about a right triangle we know of course that one of the angles is a right angle. We also know that the other two

angles are acute angles (why?). In fact we know that the other two angles are complementary angles. Therefore there is a

relationship between the sizes of the angles that the two acute angles have measures that add up to ninety degrees.What about sides? Is there a relationship between the sides of a right triangle? We know from previous lessons that if we

have the lengths of just two of the sides we can construct the triangle so it is enough to know the lengths of two sides to

determine the length of the third side. We shall now try to figure out the relationship. We shall, to make it easy to

communicate assume that the length of the hypotenuse is c units and that the two legs are of length a and b units.

So according to the Pythagorean Theorem, the area of square A, plus the area of

square B should equal the area of square C.The special sets of numbers that possess this property are called

Pythagorean Triples.The most common Pythagorean Triples are:

3, 4, 5

5, 12, 13

8, 15, 17

The Pythagorean Theorem

The Pythagorean Theoremis one of Euclidean Geometry's most beautiful theorems. It is simple, yet obscure, and is

used continuously in mathematics and physics. In short, it is really cool.

This first method is one of the ways the Pythagoreans would have proved the theorem. Unfortunately, it lacks glamour.

In the following picture let ABC be a right triangle and BD be a segment drawn perpendicular to AC.

Since the triangles are similar, the sides must be of proportional lengths.

AB/AD=AC/AB, or AB x AB = AD x ACBC/CD=AC/BC, or BC x BC= AC x CDThen, adding the two together, BC^2 + AB^2 = (AD + DC) x AC= AC^2


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Pythagorean Theorem in text book

of 10th classGiven:- ABC is a right angle Triangle

.

angle B =900

R.T.P:- AC2 = AB2 +BC2

Construction:- To draw BD AC .

A

B C

D

Proof:- In Triangles ADB and ABCAngleA=Angle A (common)Angle ADB=Angle ABC (each 900 )

ADB ~ABC ( A.A corollary )

So that AD/AB=AB/AC (In similar triangles corresponding

sides are proportional )

AB2 = AD X AC _________(1)Similarly BC2 = DCXAC _________(2)

Adding (1)&(2) we get

AB2 +BC2 = AD X AC + DCXAC

= AC (AD +DC)

= AC . AC

=AC2

Therefore AB2 +BC2 =AC2


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Typical Examples


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Example 1. Find the length of AC.

Hypotenuse

AC2 = 122 + 162 (Pythagoras Theorem)

AC2 = 144 + 256

AC2 = 400

AC = 20

A

CB

16

12Solution :


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Example 2. Find the length of diagonal d .

10

24 d

Solution:

d2 = 102 + 242 (Pythagoras Theorem)

d = +

=

=

10 24

26

2 2

676


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16km

12km

1.A car travels 16 km from east to west. Then it turns left

and travels a further 12 km.Find the displacement betweenthe starting point and the destination point of the car.

N

?

Application of Pythagoras Theorem


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16 km

12 km

A

B

C

Solution :

In the figure,

AB = 16

BC = 12

AC2 = AB2 + BC2 (Pythagoras Theorem)

AC2 = 162 + 122

AC2

= 400AC = 20

The displacement between the starting point and the

destination point of the car is 20 km


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2. The height of a tree is 5 m. The distance between

the top of it and the tip of its shadow is 13 m.

Solution:

132

= 52

+ L2

(Pythagoras Theorem)L2 = 132 - 52

L2 = 144

L = 12

Find the length of the shadow L.

5 m13 m

L


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. 1.Find the length of the hypotenuse for right triangles with legs, and sketch the triangles

a).3 and 4?

b).5 and 12?

c).5.2 and 10.5?

2.Find the lengths of the other leg of right triangles if one leg =6 and the hypotenuse =2,8.3 and 7 in

each case.

3. A right triangle with legs equal to 5cm and 12cm. What is the length of the hypotenuse? .

4. In your own words, explain what the Pythagorean Theorem states.

Exercise


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Text book of class X

Subject:MathematicsSearch engine:www.google.com

Prepared by S.Jaya Prada

coresub math att 4pythagoreantheorem

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