The following steps would be useful to prove that three points are collinear using the equation of line.
Step 1 :
Find the equation of the line using the first two points.
Step 2 :
Substitute the third point into the equation of line. If the point satisfies the equation of line, then the three points are collinear (lying on the same line).
By using the concept of equation of line, prove that the three points are collinear.
Example 1 :
(4, 2), (7, 5), (9, 7)
Solution :
Formula for slope of a line joining two points :
m = (y_{2 }- y_{1})/(x_{2 }- x_{1})
Substitute (x_{1}, y_{1}) = (4, 2) (x_{2}, y_{2}) = (7, 5).
m = (5 - 2)/(7 - 4)
= 3/3
= 1
Equation of line in slope-intercept form :
y = mx + b
Substitute m = 1.
y = x + b ----(1)
The line is passing through the point (4, 2).
So, substitute x = 4 and y = 2.
2 = 4 + b
-2 = b
Substitute b = -2 in (1).
y = x - 2
Now, substitute the third point (9, 7) into the equation of line and check whether the point satisfies the equation.
9 = 7 - 2 ?
7 = 7 ✓
The above result is true. So the point (9, 7) satisfies the equation of line.
Hence, the three points (4, 2), (7, 5) and (9, 7) are collinear.
Example 2 :
(1, 4), (3, -2), (-3, 16)
Solution :
Formula for slope of a line joining two points :
m = (y_{2 }- y_{1})/(x_{2 }- x_{1})
Substitute (x_{1}, y_{1}) = (1, 4) (x_{2}, y_{2}) = (3, -2).
m = (-2 - 4)/(3 - 1)
= -6/2
= -3
Equation of line in slope-intercept form :
y = mx + b
Substitute m = -3.
y = -3x + b ----(1)
The line is passing through the point (1, 4).
So, substitute x = 1 and y = 4.
4 = -3(1) + b
4 = -3 + b
Substitute b = 7 in (1).
y = -3x + 7
Now, substitute the third point (-3, 16) into the equation of line and check whether the point satisfies the equation.
16 = -3(-3) + 7 ?
16 = 9 + 7 ?
16 = 16 ✓
The above result is true. So the point (-3, 16) satisfies the equation of line.
Hence, the three points (1, 4), (3, -2) and (-3, 16) are collinear.
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