Show that the following sets of points are collinear.
(a) (2, 5), (4, 6) and (8, 8)
(b) (1, -1), (2, 1) and (4, 5).
Answers
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Given : (a) (2, 5), (4, 6) and (8, 8)
(b) (1, -1), (2, 1) and (4, 5).
To prove : Given points are collinear.
Solution :
(a) Let three given points be A(2, 5), B(4, 6) and C(8, 8).
Area of the triangle ∆ABC, A = ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
A = ½ |2(6 - 8) + 4(8 - 5) + 8(5 - 6)|
A = ½ |2 × - 2 + 4 × 3 + 8 × - 1|
A = ½ |- 4 + 12 - 8|
A = ½ |8 - 8|
A = ½ × 0
A = 0
Since the area of ∆ABC is zero.
Hence, the given Points A, B, C are collinear.
(b) Let three given points be A(1, −1), B(2, 1) and C(4, 5)
Area of the triangle ∆ABC, A = ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
A = ½ |1(1 – 5) + 2(5 + 1) + 4(-1 – 1)|
A = ½ |1 × - 4 + 2 × 6 + 4 × - 2|
A = ½ |- 4 + 12 - 8|
A = ½ |8 - 8|
A = ½ × 0
A = 0
Since the area of ∆ABC is zero.
Hence, the given Points A, B, C are collinear.
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Some more questions :
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
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Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
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