Phythagoras Theorem

Pythagoras Theorem 

Statement

" In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides"

Given

ABC is a right triangle. AC, AB and BCare hypotenuse and the other two sides respectively.

To Prove that

(AC)² = (AB)² + (BC)²

Construction

Draw BD perpendicular on AC.

Proof

Consider in right triangle ABC and ADB
<BAC = <BAD   (Common)
<ABC = < ADB  (Each right angle)
<ACB = <ABD   ( Remain angle)

Hence,
Triangle ABC and ADB are similar under AAA similarity.

Therefore,
AC/AB = AB/AD
ACxAD = (AB)²  ........  (i)

Now, consider in right triangle ABC and BDC
<BAC = <CBD   (Remain angle)
<ABC = < CDB  (Each right angle)

<ACB = <BCD   ( Common angle)

Hence,
Triangle ABC and BDC are similar under AAA similarity.

Therefore,
AC/BC = BC/DC
ACxDC =  (BC)² ......... (ii)

Now, add (i) and (ii)
ACxAD + ACxDC = (AB)² + (BC)²
AC(AD + DC) = (AB)² + (BC)²

From figure it is clear that
AD + DC = AC
Therefore,
ACxAC = (AB)² + (BC)²
(AC)² = (AB)² + (BC)² Proved.