Question

in a triangle ABC right angle at B.D is the mid point of BC.prove that AC²= AD²+3CD²​

Answer 1

Answer:

Given: In △ABC, ∠B = 90° and D is the mid-point of BC.

To Prove: AC2 = AD2 + 3CD2

Proof:

In △ABD,

AD2 = AB2 + BD2

AB2 = AD2 - BD2 .......(i)

In △ABC,

AC2 = AB2 + BC2

AB2 = AC2- BD2 ........(ii)    

Equating (i) and (ii)

AD2 - BD2 = AC2 - BC2

AD2 - BD2 = AC2 - (BD + DC)2

AD2 - BD2 = AC2 - BD2- DC2- 2(BDx DC )

AD2 = AC2 - DC2 - 2DC2 (DC = BD)

AD2 = AC2 - 3DC2

AC2=AD2+3CD2

Step-by-step explanation:

Answer 2

Answer:

∆ABC~∆ADB~∆BDC

<B=adb=bdc=90°

ad2+bd2=ab2••••••••••(1)

cd2+bd2=bc2 •••••••••(2)

ab2+bc2=ac2••••••••••(3)

bd2=ad×cd••••••••••••••(4)

ad=dc

adingin eq 1&2

ad2+cd2+2bd2=ab2+bc2

ad2+cd2+2(ad×dc)2=ac2

ad2+cd2+2cd2=ac2

ad2+3cd2=ac2