Question

if (1, 1) and (- 1, - 1) are the two vertices of an equilateral triangle third vertex would be​

Answer 1

Distance formula

→ √[(x2 - x1)² + (y2 - y1)²]

Now for first side of equilateral ∆.

→ √[(1 + 1)² + (1 + 1)²]

→ √(2² + 2²)

→ √8 = 2√2

For 2nd side of equilateral ∆,

→ √[(x - 1)² + (y - 1)²]

And for 3rd side of equilateral

→ √[(x + 1)² + (y + 1)²]

Since all sides of equilateral ∆ are equal.

→ 8 = (x - 1)² + (y - 1)²

→ 8 = x² + 1 - 2x + y² + 1 - 2y

→ 6 = x² + y² - 2x - 2y

Similarly,

→ 6 = x² + y² + 2x + 2y

As L.H.S of both equations are same,

→ 2x + 2y = - 2x - 2y

→ x + y = 0

→ y = - x

By substituting this value we get,

→ 6 = x² + (-x)² + 2x + 2(-x)

→ 6 = x² + x²

→ 3 = x²

→ √3 = x

Therefore y,

→ y = - √3

Coordinate : (√3, - √3)