Pythagoras theorem

 Pythagoras theorem proof with right angle triangle 

Introduction :-

Pythagoras theorem can be defined as the square of the hypotenuse of a right angle triangle is equal to the square of the base and perpendicular.

i.e. (H)^2 = (B)^2 + (P)^2

It was introduced by Greek philosopher Pythagoras. 

 

Prove of Pythagoras theorem :-

Given :-

triangle ABC right angled at D= 90 degree

to prove :-

(AB) ^2 =  (AC)^2+(CB)^2

construction :-

draw a perpendicular on base AB

Proof :-

triangle ACB ~  triangle ABC  (If a perpendicular is drawn from the vertex of the right triangle to the hypotenuse 

then triangle on both sides of the perpendicular are similar to the whole triangle and to each other.)

AD/AC = AC/AB (sides are proportional as the triangles are similar).

so,   AD.AB = AC^2 .....(1)

Similarly,

 triangle CDB ~ triangle ABC

so,

DC/CB = CB/AB

now,  DC.AB= CB^2......(2)

adding (1) and (2),

AD.AB+ DC.AB = AC^2 +CB^2

AB(AD+DC) = AC^2 +CB^2

AB.AB = AC^2 + CB^2

AB^2 = AC^2 + CB^2

Hence proved !!

 

Question on Pythagoras;-

(1) A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground . Find  the length of the ladder.

 sol)  Let AB be the ladder and CA be the wall with window at A.

also,  BC = 2.5 and CA= 6 m

 from Pythagoras theorem,we have ;

        AB^2 = BC^2 + CA^2

                   = 2.5^2 + 6^2

                   = 42.25

so,         AB  = 6.5

thus, length of the ladder is 6.5 m.