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Pythagorean Theorem


Since ancient civilizations, phythagorean theorem is very well known to us and is named by the greek mathematician Pythagoras. It is one of the most famous mathematical contribution. The theorem is of fundamental importance in Euclidean geometry as it serves as a basis for the definition of distance between two points. Anyone who has taken geometry classes in high school cannot fail to remember this, other math notions are easily thoroughly forgotten.
 It relates the lengths of the three sides of any right triangle. Sum of the two small squares equals the big one. It is a statement about triangles containing a right angle, lengths of the three sides of any right triangle.
Pythagorean Theorem

Pythagorean Theorem Formula

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Given below are some of the formulas based on Pythagoras theorem :

1) (Hypotenuse)$^{2}$ = (Base)$^{2}$ + (Perpendicular)$^{2}$
Using the above formula we can easily find the base and the perpendicular of any right triangle.

2) Hypotenuse sector $l$ and $m$ is

$l$ = $\frac{(a)^{2}}{c}$

$m$ = $\frac{(b)^{2}}{c}$

3) Perimeter of a right angled triangle is
Perimeter = $a$ + $b$ + $c$

4) Height, h = $\sqrt{(l \times m)}$

5) Surface area = $\frac{(a+b)}{2}$

Converse of the Pythagorean Theorem

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Statement : If the square of one side of a triangle is equal to the sum of the squares of the other two sides then the triangle is a right triangle.
Proof :

Given : $c^2 = a^2 + b^2$

To Prove: $\angle$ C = 90$^0$

Converse of Pythagorean Theorem

Consider one more triangle similar as above.

Given below is a $\triangle$ EGF,
Converse of Pythagorean Theorem Example

From the $\triangle$ ACB and $\triangle$ EGF

We see that

AC = EG = b and

BC = FG = a

Using Pythagoras theorem in $\triangle$ EGF, we have

(EF)$^{2}$ = (EG)$^{2}$ + (FG)$^{2}$

= b$^{2}$ + a$^{2}$                          ------1

Similarly from $\triangle$ ACB

$(AB)^{2}$ = $(AC)^{2}$ + $(BC)^{2}$  (Given)

= $b^{2}$ + $a^{2}$                          -------2

From equations 1 and 2,

$(EF)^{2}$ = $(AB)^{2}$

Therefore, $\angle$ ACB is equivalent to $\angle$ EGF.

=> $\angle$ ACB a right angle.

Thus $\triangle$ ACB is a right angled triangle.

Pythagoras theorem converse also exists and hence proved.

Pythagorean Theorem with Radicals

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Now we know what the Pythagorean theorem says, (sum of squares of sides is equal to the squares of hypotenuse of right triangles). Here we will learn how to write radical expression in simplest form. Radical is used to indicate square roots and other roots. Symbol of radical is $\sqrt{}$.

Given below is an example:
Example : For the given problem lets find the value for x.

 Pythagorean Formula
Solution :
Using Pythagorean theorem,

($\sqrt{ 24- 2x})^{2}$ = 3 $^{2}$ + x$^{2}$

24 - 2x = 9 + x$^{2}$

x$^{2}$ + 2x -15 = 0

x$^{2}$  - 3x + 5x - 15 = 0

x = 3 or x = - 5

Value of x = - 5 is neglected and only positive value is considered.

Therefore, for the given problem value of x is 3.

Pythagorean Theorem Examples

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Example 1 : For a right angled triangle base is 10 cm and the perpendicular is 12cm. Calculate the length of the hypotenuse?

Solution : Given : Length of the perpendicular : 10 cm

Formula to find the length of the hypotenuse is given below.

Hypotenuse = $\sqrt{(Base)^{2} + (Perpendicular)^{2}}$

= $\sqrt{(10)^{2} + (12)^{2}}$

= $\sqrt{100 + 144}$

= $\sqrt{244}$

= 15.62

Example 2 : For a right angled triangle if the hypotenuse is 15cm and the perpendicular is 3cm. Find the length of the base of the triangle?

Solution : Using Pythagoras theorem,

h$^{2}$  = p$^{2}$ + b$^{2}$

15$^{2}$  = 3$^{2}$ + b$^{2}$

144 = b$^{2}$

b = 12

Base is 12 cm