Question

in an triangle ABC altitude AD divides BC in the ratio 2:3 prove that 5(ACsquare -ABsquare)=BCsquare.

Answer 1

BD/DC=2/3⇒3 BD=2 DC⇒BD=2/3 DC
In right ΔADC,AC²=AD²+DC²
In rightΔADB,AB²=AD²+BD²=AD²+(2/3 DC)²=AD²+4/9 DC²
AB²=AD²+4/9 DC²
5(AC²-AB²)=5(AD²+DC²-AD²-4/9 DC²)=5(5/9 DC²)=25/9 DC²
5(AC²-AB²)=25/9 DC².....(1)
BC²=(BD+DC)²
=(2/3 DC+DC)²=(5/3 DC)²=25/9 DC²
BC²=25/9 DC².....(2)
From (1) and (2)
5(AC²-AB²)=BC²..........Proved