Question

Prove that the area of a triangle with vertices (t, t -2), (t +2, t + 2) and(t + 3, t) is independent of t.

Answer 1

let the points be `(x_1, y_1), (x_2, y_2) , (x_3, y_3)`
Area of triangle ` = 1/2 |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|`
now, area of triangle is `= 1/2|t(t+2-t)+(t+2)(t-t+2) + (t+3)(t-2-t-3)|`
`=1/2|t(2) + 2(t+2) - 4(t+3)|`
`= 1/2|2t + 2t + 4 - 4t - 12|`
`= 1/2|4 -12|`
`= 1/2|-8| = 8/2 = 4`
area of triangle= `4`
`:.` independent of t
hence proved