Determine whether the points are collinear. A(1, -3), B(2, -5), C(-4, 7)
aNSWER
By distance formula,
A(1, –3), B(2, –5), C(–4, 7)
[ =d(A, B) sqrt{((2 – 1)^2 + [-5 – (-3)]^2)} ]
[ ∴ d(A, B)= sqrt{((1)^2 + (-5 + 3)^2)}]
[∴ d(A, B)= sqrt{(1)^2 + (-2)^2)}]
[∴ d(A, B) = sqrt{(1+ 4)}]
[ ∴ d(A, B) = sqrt{(5)} …(1)]
[d(B, C) = sqrt((- 4 – 2)^2 + [7 – (-5)]^2) ]
[ ∴ d(B, C) = sqrt((-6)^2 + [7 + 5]^2) ]
[∴ d(B, C) = sqrt((-6)^2 + (12)^2) ]
[∴ d(B, C) = sqrt(36 + 144) ]
[∴ d(B, C) = sqrt(180) ]
[ ∴ d(B, C) = sqrt(36 × 5) ]
[ ∴ d(B, C) =6 sqrt(5) …(2) ]
[d(A, C) = sqrt([-4 – 1]^2 + [7 – (-3)]^2) ]
[{ ∴ d(A, C) = sqrt((-4 – 1)^2 + (7 + 3)^2)} ]
[∴ d(A, C) = sqrt((-5)^2 + (10)^2) ]
[∴ d(A, C) = sqrt(25 + 100) ]
[∴ d(A, C) = sqrt(125) ]
[∴ d(A, C) = sqrt(25 × 5) ]
[∴ d(A, C) = 5 sqrt{(5)}` …(3)]
Adding (1) and (3)
∴ d(A, B) + d(A, C) = d(B, C)
∴ √5 + 5√5 = 6√5 …(4)
∴ d(A, B) + d(A, C) = d(B, C) …[From (2) and (4)]
∴ Points A,B and C are collinear.
Points A(1, –3), B(2, –5), and C(–4, 7) are collinear.
Chapter 5. Co-ordinate Geometry – Practice Set 5.1 (Page 107)