Yash's pythogoras theorem ppt.Class X

17
A PRESNTATION ON GOLDEN RATIO

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Transcript of Yash's pythogoras theorem ppt.Class X

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A PRESNTATION ON

GOLDEN RATIO

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Michael Mastlin

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A LITTLE MOREIn any right triangle, the square of the length of the hypotenuse is equal to the sum of 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

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Starting with a right triangle and squares on each side, the middle size square is cut into congruent quadrilaterals .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.

Animated Proof of the Pythagorean Theorem

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ANIMATED PROOF OF THE PYTHAGOREAN THEOREM

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Given:- ABC is a right angle Triangle. angle B

=900

T.P:- AC2 = AB2 +BC2

Construction:- To draw BD AC .

A

B C

D

Proof:- In ADB and ABC Angle A = Angle A (common) Angle ADB = Angle ABC (each 900 ) ADB ~ ABC ( A.A corollary ) So that AD/AB = AB/AC

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

PYTHAGOREAN THEOREM IN TEXT BOOK OF 10TH CLASS

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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 + 256AC2 = 400AC = 20

A

CB

16

12

Solution :

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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 between the starting point and the destination point of the car.

N

?

Application of Pythagoras’ Theorem

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16 km

12 km

AB

C

Solution :

In the figure,AB = 16kmBC = 12km

AC2 = AB2 + BC2 (Pythagoras’ Theorem)AC2 = 162 + 122

AC2 = 400AC = 20km

The displacement between the starting point and the destination point of the car is 20 km.

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Q.) The height of a tree is 5 m. The distance between the top of it and the tip of its shadow is 13 m. Find the length of the shadow L.

Solution:132 = 52 + L2 (Pythagoras’ Theorem)L2 = 132 - 52

L2 = 144L = 12

5 m13 m

L

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All efforts by yash agarwal


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