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Question
If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x1, y1), (x2, y2), (x3, y3), then `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`
Solution
The area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by
Δ = `1/2 |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|`
Also, area of an equilateral triangle with side a is given by
Δ = `sqrt(3)/2 "a"^2`
∴ `1/2 |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)| = sqrt(3)/4 "a"^2`
Squaring both sides, we get
⇒ Δ2 = `1/4 |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)| = 3/16 "a"^4`
or `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`
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