How can we use the concept: To determine if points are collinear, create three pairs and determine if any two of the three pairs share the same slope. If so, they are collinear. Use the slope formula to determine the slopes.
Example: Determine if the following points are collinear:
Points: C (0,2) R (3,5) M (6,8)
Pairs formed: CR, CM, and RM
Slope Formula: (y₂ - y₁) / (x₂ - x₁)
Pair CR slope: (5-2) / (3-0)= 3/3 = 1
Pair CM slope: (8-2) / (6-0)= 6/6 = 1
Pair RM slope: (8-5) / (6-3)= 3/3 = 1
Any combinations of two pairs share the same slope, so the points are collinear.
Sample Math Problems
Question
Determine if the following points are collinear:
Points: C (10,12) R (10,14) M (11,14)
Answer
Pairs formed: CR, CM, and RM
Slope Formula: (y₂ - y₁) / (x₂ - x₁)
Pair CR slope: (14-12) / (10-10)= 2/0 = undefined
Pair CM slope: (14-12) / (11-10)= 2/1 = 2
Pair RM slope: (14-14) / (11-10)= 0/1 = 0
Solution: None share slopes so they are not collinear points
Question
Determine if the following points are collinear:
Points: T (7,8) R (8,9) S (9,10)
Answer
Pairs formed: TR, TS, and RS
Slope Formula: (y₂ - y₁) / (x₂ - x₁)
Pair TR slope: (9-8) / (8-7)= 1/1 = 1
Pair TS slope: (10-8) / (9-7)= 2/2 = 1
Pair RS slope: (10-9) / (9-8)= 1/1 = 1
Solution: All pairs share the same slope so they are collinear points
Question
Determine if the following points are collinear:
Points: A (7,9) B (10,12) C (16,18)
Answer
Pairs formed: AB, AC, and BC
Slope Formula: (y₂ - y₁) / (x₂ - x₁)
Pair AB slope: (12-3) / (10-7)= 9/3 = 3
Pair AC slope: (18-9) / (16-7)= 9/9 = 1
Pair BC slope: (18-12) / (16-10)= 6/6 = 1
Solution: None share slopes so they are not collinear points
Question
Determine if the following points are collinear:
Points: L (3,3) W (5,5) K (7,7)
Solution
Pairs formed: LW, LK, and WK
Slope Formula: (y₂ - y₁) / (x₂ - x₁)
Pair LW slope: (5-3) / (5-3)= 2/2 = 1
Pair LK slope: (7-3) / (7-3)= 4/4 = 1
Pair WK slope: (7-5) / (7-5)= 2/2 = 1
Solution: All pairs share the same slope so they are collinear points
