Question

if triangle ABC is right angled at B And M is a point on AC such that BM is perpendicular to AC prove AB² =AM×BM​

Answer 1

$\: \: Given :-$

if triangle ABC is right angled at B And M is a point on AC such that BM is perpendicular to AC prove AB² =AM×BM

$\: \: Solution :-$

The pictorial form of the given problem is as follows,↗️

Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

First, we consider the ΔABM and applying Pythagoras theorem we get,

AM2 = AB  + BM2 

AB2 = AM2 - BM2               ......(i)

Now, we consider the ΔABC and applying Pythagoras theorem we get,

AC2 = AB2  + BC2 

AB2 = AC2 - BC2                ......(ii)

From (i) and (ii) we get,

AM2  - BM2  = AC2  - BC2 

AM2  + BC2 = AC2 + BM2  

Hence Proved.

Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.