Question

If E is a point on side CA of an equilateral triangle ABC such that BE ⊥ CA, then AB² + BC² + CA² =

Options

• 2 BE²

• 3 BE²

• 4 BE²

• 6 BE²

Answer 1

In triangle ABC, E is a point on AC such that  `BE ⊥ AC`.

We need to find `AB^2+BC^2+AC^2`.

Since `BE ⊥ AC`, CE = AE =

\[\frac{AC}{2}\](In a equilateral triangle, the perpendicular from the vertex bisects the base.)

In triangle ABE, we have

`AB^2 = BE^2+AE^2`

Since AB = BC = AC

Therefore, `AB^2=BC^2=AC^2=BE^2+AE^2`

Since in triangle BE is an altitude, so  `BE = (sqrt3)/2 AB`

`BE = (sqrt3)/2 AB`

`(sqrt3)/2 xxAC`

`(sqrt3)/2 xx 2AE= sqrt3AE`

`⇒ AB^2 + BC^2+AC^2= 3BE^2+3((BE)/sqrt3)^2`

`= 3BE^2+BE^2=4BE^2`

Hence option (c) is correct.