Question

triangle ABC is right-angled triangle ,angle B=90°point Dr is on side BC then prove that BC^2+AD^2=AC^2+BD^2

Answer 1

Given that Triange ABC with B having angle 90 degrees.

D is any point on BC.

We find that now ABD is also a right triangle right angled at B.

Hence we can use Pythagorean theorem to prove the given.

Consider triangle ABC, with hypotenuse at AC.$AC^2 = AB^2 +AC^2 :</p><p>AB^2 = AC^2-BC^2$

Similarly for triangle ACD, with hypotenuse at AD.$AD^2 = AB^2 +BD^2 :</p><p>AB^2 = AD^2-BD^2$

It follows that

$AD^2-BD^2 = AC^2-BC^2 Or</p><p>BC^2+AD^2 = AC^2+BD^2$