Question

Find the area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).

Answer 1

Let A = (1, 1, 0), B = (1, 0, 1), C = (0, 1, 1)

l(AB) = `sqrt((1 - 1)^2 + (1 - 0)^2 + (0 - 1)^2) = sqrt(0 + 1 + 1) = sqrt2`

l(BC) = `sqrt((1 - 0)^2 + (0 - 1)^2 + (1 - 1)^2) = sqrt(1 + 1 + 0) = sqrt2`

l(CA) = `sqrt((0 - 1)^2 + (1 - 1)^2 + (1 - 0)^2) = sqrt(1 + 0 + 1) = sqrt2`

∴ l(AB) = l(BC) = l(CA)

∴ Triangle is an equilateral triangle

∴ Its area = `sqrt 3/4 ("side")^2`

= `sqrt 3/4 (sqrt2)^2`

= `sqrt3/2` sq. units.