In ΔABC, m ∠ B = 90 and is a median. Prove that AB 2 + BC 2 + AC 2 = 8BE 2 .

Question

Philoid · Accepted Answer

mB = 90

is a median

∴ AE = EC

⇒ AC = 2AE

Applying Pythagoras Theorem :

AB² + BC² = AC² …Equation (i)

Applying Appoloneous theorem

AB² + BC² = 2(AE² + BE²)

⇒ AC² = 2(AE² + BE²)

Now since AC = 2AE

⇒ (2AE)² = 2(AE² + BE²)

⇒ 2AE² = 2BE²

⇒ AE = BE …Equation (ii)

∴ AB² + BC² + AC² = 2AC² (From Equation (i))

⇒ AB² + BC² + AC² = 2 × (2AE)² = 8AE²

From Equation (ii) we can say that

⇒ AB² + BC² + AC² = 8BE²

In ΔABC, m∠B = 90 and is a median. Prove that AB2 + BC2 + AC2 = 8BE2.