Question

In a right angle ABC if D and E trisect BC then prove that 8AE^2= 3AC^2+5AD^2​

Answer 1

Let BC = DE = EC = x

BC = 3x

In Δ ABD,

AD² = AB² + BD²

AD² = AB² + x²

In Δ ABE,

AE² = AB² + BE²

AE² = AB² + (2x)²

AE² = AB² + 4x²

In Δ ABC,

AC² = AB² + BC²

AC² = AB + (3x)²

AC² = AB² + 9x²

Now,

3AC² + 5AD² = 3(AB² + 9x²) + 5(AB² + x²)

8AB² + 32x²

8(AB² + 4x²)

= 8AE²

⇒ 8AE² = 3AC² + 5AD²

Hence proved.

though you understand