Question

in triangle Abc angle b=90 degree and d is the midpoint of bc prove that ac^2=ad^2+3cd^2

Answer 1

Hope this Helps..

The Answer is attached as photograph..

Answer 2

Answer:

$AC^2=AD^2+3CD^2$

Step-by-step explanation:

Given information: ΔABC is a right angled triangle, ∠B=90° and D is the midpoint of BC.

To prove : $AC^2=AD^2+3CD^2$

Proof :

According to the Pythagoras theorem, in a right angles triangle

$perpendicular^2+base^2=hypotenuse^2$

Use Pythagoras theorem in triangle ABC.

$(AB)^2+(BC)^2=(AC)^2$

$(AB)^2=(AC)^2-(BC)^2$                   ..... (1)

Use Pythagoras theorem in triangle ABD.

$(AB)^2+(BD)^2=(AD)^2$

$(AB)^2=(AD)^2-(BD)^2$                   ..... (2)

Equating (1) and (2), we get

$(AC)^2-(BC)^2=(AD)^2-(BD)^2$

Add $(BC)^2$ on both sides.

$(AC)^2=(AD)^2-(BD)^2+(BC)^2$

D is the midpoint of BC,

$(AC)^2=(AD)^2-(CD)^2+(BC)^2$                (BD=CD)

$(AC)^2=(AD)^2-(CD)^2+(2CD)^2$            (BC=2CD)

On further simplification, we get

$(AC)^2=(AD)^2-(CD)^2+4(CD)^2$

$(AC)^2=(AD)^2+3(CD)^2$

Hence proved.