SOLUTION: Are (2,350) and (5,200) and (6,150) collinear?

Question 1203348: Are (2,350) and (5,200) and (6,150) collinear?
Found 3 solutions by josgarithmetic, math_tutor2020, greenestamps:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
The ordered pair, the three of them, if collinear, the slopes of each PAIR of ordered pair would be the same. You undoubtedly know what to do now.
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I'll go over the basic outline of four approaches.

For each approach
A = (2,350)
B = (5,200)
C = (6,150)

Approach 1:
Find slopes of line AB and line BC.
If the slopes are equal, then the 3 points fall on the same line (to make them collinear).
As optional practice you can find the slope of line AC, but it's not required.

Approach 2:
Determine the equation of line AB and call this function f.
If f(6) = 150, then C is also on line AB to make the 3 points collinear.

Approach 3:
Focus on the x coordinates to see that because 2 < 5 < 6, it must mean B is horizontally between A and C.
This in turn then would mean AB+BC = AC if and only if A,B,C are on the same line together.
Use the distance formula to calculate the lengths of AB, BC, and AC.

Approach 4:
Use the distance formula to calculate the lengths of AB, BC, and AC.
Afterward, use Heron's formula to find the area of triangle ABC.
If area = 0 then the points are collinear because we have a degenerate triangle aka a straight line.
If area > 0 then the points are not collinear because an actual triangle forms.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

Looking at the x coordinates of the three points, we see B is between A and C. So the three points are collinear if the slope from A to B is the same as the slope from B to C.

Informally, you can see the slopes are the same by seeing that the change in the x value from A to B is 3 times the change in the x value from B to C while the change in y from A to B is also 3 times the change in the y value from B to C.

Using the given coordinates, the x value changes by 3 from A to B and changes by 1 from B to C; the change in x from A to B is 3 times the change in x from B to C.

The three points are collinear if the change in y from A to B is 3 times the change in y from B to C. From A to B the change in y is -150 and from B to C the change in y is -50; -150 is 3 times -50, so the points are collinear.

This is an informal method that shows the points are collinear by seeing that the slope from A to B and the slope from B to C are the same -- but without using the formal formula for determining the slopes.

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