Question

In a triangle ABC, N is a point on AC such that BN ⊥ AC. If BN² = AN . NC, prove that ∠B = 90°.

Answer 1

In  ΔABC , BN ⊥ AC.

Also, `BN^2=ANxxNC`

We have to prove that  `∠ B= 90^o`.

In triangles ABN and BNC, we have

`AB^2=AN^2+BN^2`

`BC^2=BN^2+CN^2`

Adding above two equations, we get

`AB^2+BC^2=AN^2+CN^2+2BN^2`

Since `BN^2= AN.NC`

So,

`AB^2+BC^2=AN^2+CN^2+2BN^2`

`AB^2+BC^2=(AN+NC)^2`

`AB^2+BC^2=AC^2`

Hence `∠ B = 90^o`