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Question
Determine whether the points are collinear.
A(1, −3), B(2, −5), C(−4, 7)
Solution
A(1, −3), B(2, −5), C(−4, 7)
Let,
A(1, −3) = A(x1, y1)
B(2, −5) = B(x2, y2)
C(−4, 7) = C(x3, y3)
By distance formula,
d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
= `sqrt((2 - 1)^2 + [-5 - (-3)]^2)`
= `sqrt((1)^2 + (-5 + 3)^2)`
= `sqrt((1)^2 + (-2)^2)`
= `sqrt(1+ 4)`
= `sqrt(5)` ...(1)
d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2)`
= `sqrt((- 4 - 2)^2 + [7 - (-5)]^2)`
= `sqrt((-6)^2 + [7 + 5]^2)`
= `sqrt((-6)^2 + (12)^2)`
= `sqrt(36 + 144)`
= `sqrt(180)`
= `sqrt(36 × 5)`
= `6sqrt(5)` ...(2)
d(A, C) = `sqrt((x_3 - x_1)^2 + (y_3 - y_1)^2)`
= `sqrt((-4 - 1)^2 + [7 - (-3)]^2)`
= `sqrt((-4 - 1)^2 + (7 + 3)^2)`
= `sqrt((-5)^2 + (10)^2)`
= `sqrt(25 + 100)`
= `sqrt(125)`
= `sqrt(25 × 5)`
= `5sqrt(5)` ...(3)
Adding (1) and (3)
∴ d(A, B) + d(A, C) = d(B, C)
∴ `sqrt5 + 5sqrt5 = 6sqrt5` ...(4)
∴ d(A, B) + d(A, C) = d(B, C) ...[From (2) and (4)]
∴ Points A(1, −3), B(2, −5) and C(−4, 7) are collinear.
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