Collinearity of three points
Core Concept
WHY? A line is1-dimensional—it has length but no width. If three points are collinear, they can't enclose any 2D region, so the "triangle" they form is degenerate (collapsed).
The Area Method: Deriving from First Principles
WHAT we're doing: Using the area formula for a triangle to detect collinearity.
Starting point: The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is:
WHY this formula? It comes from the determinant method (cross product in 2D). If we place vectors from A to B and A to C, their cross product gives twice the area.
Derivation:
- Vector AB = (x₂ - x₁, y₂ - y₁)
- Vector AC = (x₃ - x₁, y₃ - y₁)
- Area = ½|AB × AC| = ½|(x₂ - x₁)(y₃ - y₁) - (x₃ - x₁)(y₂ - y₁)|
- Expanding: ½|x₂y₃ - x₂y₁ - x₁y₃ + x₁y₁ - x₃y₂ + x₃y₁ + x₁y₂ - x₁y₁|
- Regrouping: ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
WHY the absolute value disappears? For collinearity, we need Area = 0, so the expression inside must equal zero. The absolute value is irrelevant for zero.
The Slope Method: Alternative Approach
WHAT: If three points are collinear, the slope between any two pairs must be equal.
WHY? A line has constant slope everywhere. If slope(AB) = slope(BC), then B doesn't cause a direction change.
Cross-multiplying (to avoid division by zero): $(y_2 - y_1)(x_3 - x_2) = (y_3 - y_2)(x_2 - x_1)$$
\text{Area} = \frac{1}{2} |x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|
WHY this step? We substitute coordinates directly into the formula. Result: Yes, colinear! (The expression equals zero.)
Verification (Slope Method):
- Slope AB = (4-2)/(3-1) = 2/2 = 1
- Slope BC = (6-4)/(5-3) = 2/2 = 1
- Equal slopes → collinear ✓
Solution:
WHY this matters? Non-zero result means the triangle has area =½|-2| = 1, so points form a real triangle. Result: NOT collinear.
Solution:
WHY this step? We treat k as unknown and solve the colinearity equation. Verification: Slope AB = (5-3)/(4-2) = 1, Slope BC = (7-5)/(6-4) = 1 ✓
## Common Mistakes
Why it feels right: The area formula has absolute value, so students include it.
The fix: For checking collinearity, we need the expression to equal zero. |anything| = 0 means the inside is zero, so we can drop the absolute value immediately. Including it doesn't break the math but is redundant.
Why it feels right: The cyclic pattern looks similar, easy to mix up.
The fix: The formula is cyclic but specific: x₁(y₂ - y₃), x₂(y₃ - y₁), x₃(y₁ - y₂). Each x-coordinate multiplies the difference of the OTHER two y-coordinates. Mnemonic: "1 skips itself, multiplies 2 and 3."
Why it feels right: Slope method is simpler, students reach for it first.
The fix: If any two x-coordinates are equal, check if ALL three x-coordinates are equal (vertical line case). Otherwise, use the area method which always works.
Active Recall Practice
#flashcards/maths
What is the condition for three points to be collinear? :: Three points A(x₁,y₁), B(x₂,y₂), C(x₃,y₃) are collinear if x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) = 0
Why does colinearity mean zero area?
What's the slope condition for collinearity?
When does the slope method fail?
If points A(1,2), B(3,k), C(5,6) are collinear, what is k?
What does a non-zero area value tell you?
Feynman Technique
Recall Explain to a 12-year-old
Imagine you're walking from your house (point A) to your friend's house (point B) to the park (point C). If all three places are on the same street and you never have to turn, they're collinear—on the same line!
Now, if you DO have to turn at your friend's house to get to the park, the three places form a triangle. That triangle takes up space on the map.
Here's the trick: mathematicians measure how much space (area) that triangle takes. If the area is ZERO, it means you never turned—all three points are one straight line!
We use a magic formula with the coordinates (like addresses on a map). Plug in the numbers, do the math, and if you get zero, boom—they're collinear! If you get any other number, they're not on the same line.
Memory Aid
Each x-coordinate "skips itself" and multiplies the difference of the other two y-coordinates. The pattern is cyclic: (2-3), (3-1), (1-2).
Connections
- Area of Triangle using Coordinates - The parent formula from which colinearity is derived
- Slope of a Line - The alternative method for checking collinearity
- Section Formula - Used to find points on a line segment, related to collinearity
- Determinants - The area formula is actually a 3×3 determinant
- Vectors and Cross Product - The geometric interpretation in 2D
- Parametric Equations - Another way to express colinear points: all lie on r = a + t(b-a)
- Linear Dependence - In linear algebra, collinear vectors are linearly dependent
Study Tip: Practice by plotting random points and guessing if they're collinear BEFORE calculating. Build geometric intuition alongside algebraic skill. The 80/20: Master the area method (handles 100% of cases), use slope method for quick mental checks.
Concept Map
Hinglish (regional understanding)
Intuition Hinglish mein samjho
Teen points colinear tab hote hain jab wo ek hi seedhi line par ho. Socho ki tum school se ghar ja rahe ho aur bech mein ek dukaan hai—agar teno jagah ek hi road par hain aur tumhe kabhi direction change nahi karni padti, to wo colinear hain!
Mathematically check karne ke liye humein ek formula use karta hai jo triangle ka area calculate karta hai. Agar wo area zero aya, matlab teen points ek line par hain (kyunki line 2D space occupy nahi karti). Formula hai: x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) = 0. Har x-coordinate apne ap ko "skip" karta hai aur baki do y-coordinates ka difference multiply karta hai.
Dosra tarika hai slope method—agar point A se B ka slope aur B se C ka slope equal hain, to points colinear hain. Lekin yeh method vertical lines ke liye fail ho jata hai (jab x-coordinates equal ho). Area method hamesha kaam karta hai, isliye wo better hai exams mein use karne ke liye. Practice karo different points ke sath, aur dekho ki kab collinear bante hain aur kab triangle banta hai!