Fun with mathematics

Pythagoras theorem with different statement

Pythagoras theorem states that if square of a number is equal

to sum of squares of two other numbers ,then these three

numbers represent the three sides of a right-angled triangle.

Conversely, if (x,y,z) denotes the length of opposite side,

adjacent side and hypotenuse of a right-angled triangle

respectively, then

                           x^2 + y^2 = z^2

Where z>x and z>y but x+y >z.(3,4,5) and (5,12,13)  are the

two well-known examples for Pythagorean triples.

                          3^2 + 4^2 = 5^2 ;

                          5^2 + 12^2 = 13^2

 Since x^2 represents a square of a side x, we usually consider

 squares in each side of a right-angled triangle  to show that

the area of the square drawn on the hypotenuse is equal to the

sum of the areas of the squares drawn on other two sides.

It is curious to know that the Pythagoras theorem holds good

for many geometrical figures such as rectangle, circle,

semi-circle, equilateral triangle,

isosceles triangle etc.,

Squares need one variable- the side length, which is taken

as the length of each side of the given right-angled triangle.

But rectangle or isosceles triangle requires two variables,

the length and breadth  or base  and height.  The length of the

side of the triangle is taken as one of the variables for the

rectangles/isosceles triangles to be drawn on each side of the

triangle. The other variable required is derived from the other

sides of the given right-angled triangle . For example, the

length of the rectangle or the height of the isosceles triangle

drawn on the hypotenuse  is equal to the sum of the length

of the other two sides of the triangle  and for  other sides 

 it is the length of the hypotenuse.  It can be shown that the

areas of the rectangle/isosceles triangle drawn on hypotenuse

is equal to the sum of areas of the rectangles/isosceles

triangles drawn on other two sides.

               For rectangle :     xz + yz = z(x+y)

              For isosceles triangle :  ½ xz + ½ yz = ½ z(x+y)

                                                           Or xz + yz = z(x+y)

 If the length of the rectangle is taken as its breadth which is

 equal to the length of the side of the triangle over which it is

 drawn, it becomes a square. It holds good even in the case

of isosceles triangle. If the height of the isosceles triangle

is equal to the length of the side of the given right-angled triangle,

their areas make the very same Pythagoras expression

x^2+y^2= z^2.

If semi-circles/circles are drawn by keeping the sides of the

given right-angled triangle as its diameter, the n the area of

 the semi-circle/circle drawn on hypotenuse will be equal to

the sum of the areas of the semi-circles/circles drawn on

other two sides.