Collinearity Using Distance Formula
Collinear points are points that lie on the same straight line. Determining whether three given points are collinear is a fundamental problem in Chapter 7 (Coordinate Geometry) of the NCERT Class 10 textbook.
There are two primary methods to check collinearity:
- Distance formula method: Check if the sum of two shorter distances equals the longest distance.
- Area method: Check if the area of the triangle formed by the three points is zero.
This topic is closely related to the distance formula and the area of a triangle using coordinates.
The concept of collinearity is fundamental in both pure geometry and coordinate geometry. Testing whether three points are collinear is equivalent to checking whether they form a degenerate triangle (a triangle with zero area). This connection between collinearity and area provides one of the most elegant and computationally efficient methods for the test.
In CBSE Class 10 examinations, collinearity problems appear in multiple forms: verifying collinearity given specific coordinates, finding unknown values that make three points collinear, and proving general collinearity for parametric points. These problems typically carry 2-3 marks and require clear working.
Three methods exist for testing collinearity: the distance formula method, the area method, and the slope method. Each has its advantages depending on the problem type. The area method is generally the most efficient for numerical problems, while the distance method provides additional geometric insight by identifying which point lies between the other two.
What is Collinearity Using Distance Formula - Methods, Examples & Practice?
Definition: Three or more points are said to be collinear if they all lie on the same straight line.
Collinearity condition using distance formula:
Three points A, B, C are collinear if and only if AB + BC = AC
(where AC is the greatest distance among AB, BC, and AC)
Key points:
- The point B lies between A and C on the line.
- If AB + BC > AC, the points form a triangle (not collinear).
- If AB + BC = AC, the points are collinear.
- This is essentially the triangle inequality reaching equality.
Collinearity Using Distance Formula Formula
Distance Formula:
Distance between (x1, y1) and (x2, y2) =
sqrt((x2 - x1)^2 + (y2 - y1)^2)
Collinearity Condition (Distance Method):
For three points A, B, C, compute AB, BC, and AC. The points are collinear if:
AB + BC = AC (or whichever sum gives the longest side)
Collinearity Condition (Area Method):
x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) = 0
Derivation and Proof
Why the distance method works:
- If three points A, B, C are collinear with B between A and C, then B lies on segment AC.
- The distance from A to C via B equals AB + BC.
- The direct distance from A to C is AC.
- Since B is on segment AC, the path through B equals the direct distance: AB + BC = AC.
- If B is NOT on segment AC, then by the triangle inequality, AB + BC > AC.
Why the area method works:
- Three collinear points cannot form a triangle (they form a degenerate triangle).
- A degenerate triangle has zero area.
- Therefore, the area formula yields zero for collinear points.
Connection between the three methods:
- All three methods are mathematically equivalent — they give the same answer for any set of three points.
- The distance method is based on the triangle inequality: if the longest side equals the sum of the other two, the triangle degenerates to a line.
- The area method checks whether the three points form a triangle with zero area.
- The slope method checks whether the direction from A to B is the same as from B to C.
When each method is preferred:
- Distance method: When you also need to determine the order of points on the line, or when the problem involves distances.
- Area method: For most numerical and algebraic problems — it avoids square roots.
- Slope method: When the points have simple integer coordinates — quick computation. Fails for vertical lines.
Types and Properties
Types of collinearity problems:
| Problem Type | Method |
|---|---|
| Check if 3 points are collinear | Distance method or area method |
| Find unknown coordinate for collinearity | Set area = 0 or use distance condition, solve for variable |
| Prove collinearity algebraically | Substitute general expressions, simplify to show condition holds |
| Determine if point lies on a line segment | Check collinearity AND verify the point lies between the endpoints |
Important edge cases:
- If two of the three points coincide (are the same point), the three points are always collinear. Any two points define a line, and the third point (which equals one of the first two) lies on that line.
- If all three points are the same, they are trivially collinear.
- When using the slope method with vertical points (same x-coordinate), the slope is undefined. In this case, check if all three x-coordinates are equal — if so, the points are collinear (they all lie on the vertical line x = constant).
Methods
Method 1: Distance Formula Method
- Compute all three distances: AB, BC, AC.
- Identify the largest distance.
- Check if the sum of the other two equals the largest.
- If yes, the points are collinear.
Method 2: Area Method
- Use the area formula: (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
- If the area = 0, the points are collinear.
Method 3: Slope Method
- Calculate slope of AB = (y2-y1)/(x2-x1)
- Calculate slope of BC = (y3-y2)/(x3-x2)
- If slope of AB = slope of BC, the points are collinear.
Which method to use:
- For numerical problems: distance method or area method (both work well).
- For proving general collinearity: area method is usually simplest.
- For finding unknown coordinates: area method (gives a linear equation).
Detailed comparison of the three methods:
| Method | Advantages | Disadvantages |
|---|---|---|
| Distance Formula | Tells which point is between the others; geometric intuition | Requires computing square roots; more arithmetic |
| Area Method | No square roots; efficient; works for all cases | Doesn't tell the order of points on the line |
| Slope Method | Quickest for integer coordinates | Fails for vertical lines (undefined slope) |
Solved Examples
Example 1: Checking Collinearity Using Distance Formula
Problem: Check whether the points A(1, 5), B(2, 3), C(4, -1) are collinear.
Solution:
Step 1: AB = sqrt((2-1)^2 + (3-5)^2) = sqrt(1 + 4) = sqrt(5)
Step 2: BC = sqrt((4-2)^2 + (-1-3)^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5)
Step 3: AC = sqrt((4-1)^2 + (-1-5)^2) = sqrt(9 + 36) = sqrt(45) = 3sqrt(5)
Step 4: AB + BC = sqrt(5) + 2sqrt(5) = 3sqrt(5) = AC
Since AB + BC = AC, the points are collinear.
Answer: Yes, the points are collinear.
Example 2: Points NOT Collinear
Problem: Check whether A(1, 1), B(3, 4), C(5, 6) are collinear.
Solution:
Step 1: AB = sqrt((3-1)^2 + (4-1)^2) = sqrt(4 + 9) = sqrt(13)
Step 2: BC = sqrt((5-3)^2 + (6-4)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2)
Step 3: AC = sqrt((5-1)^2 + (6-1)^2) = sqrt(16 + 25) = sqrt(41)
Step 4: AB + BC = sqrt(13) + 2sqrt(2) = 3.606 + 2.828 = 6.434
AC = sqrt(41) = 6.403
Since AB + BC (6.434) is not equal to AC (6.403), the points are NOT collinear.
Answer: No, the points are not collinear.
Example 3: Using the Area Method
Problem: Check collinearity of points (2, 4), (4, 6), (6, 8) using the area method.
Solution:
- Area = (1/2)|2(6 - 8) + 4(8 - 4) + 6(4 - 6)|
- = (1/2)|2(-2) + 4(4) + 6(-2)|
- = (1/2)|-4 + 16 - 12|
- = (1/2)|0| = 0
Since the area = 0, the points are collinear.
Answer: Yes, the points are collinear.
Example 4: Finding Unknown Coordinate for Collinearity
Problem: Find the value of k for which the points (7, -2), (5, 1), (3, k) are collinear.
Solution:
Using area method (set area = 0):
- 7(1 - k) + 5(k - (-2)) + 3((-2) - 1) = 0
- 7 - 7k + 5k + 10 + 3(-3) = 0
- 7 - 7k + 5k + 10 - 9 = 0
- -2k + 8 = 0
- k = 4
Verification: Points (7,-2), (5,1), (3,4). Slope of first pair = (1+2)/(5-7) = -3/2. Slope of second pair = (4-1)/(3-5) = -3/2. Equal slopes confirm collinearity.
Answer: k = 4.
Example 5: Collinearity with Negative Coordinates
Problem: Are the points (-2, -3), (1, 0), (4, 3) collinear?
Solution:
Using distance formula:
- AB = sqrt((1+2)^2 + (0+3)^2) = sqrt(9+9) = sqrt(18) = 3sqrt(2)
- BC = sqrt((4-1)^2 + (3-0)^2) = sqrt(9+9) = sqrt(18) = 3sqrt(2)
- AC = sqrt((4+2)^2 + (3+3)^2) = sqrt(36+36) = sqrt(72) = 6sqrt(2)
AB + BC = 3sqrt(2) + 3sqrt(2) = 6sqrt(2) = AC
Since AB + BC = AC, the points are collinear.
Answer: Yes, the points are collinear.
Example 6: Using the Slope Method
Problem: Check whether the points (0, 0), (3, 6), (5, 10) are collinear using slopes.
Solution:
- Slope of AB = (6 - 0)/(3 - 0) = 6/3 = 2
- Slope of BC = (10 - 6)/(5 - 3) = 4/2 = 2
Since slope of AB = slope of BC = 2, the points are collinear.
Answer: Yes, the points are collinear.
Example 7: Proving Collinearity with General Coordinates
Problem: Prove that the points (a, b+c), (b, c+a), (c, a+b) are always collinear.
Solution:
Using area method:
- Area = (1/2)|a((c+a)-(a+b)) + b((a+b)-(b+c)) + c((b+c)-(c+a))|
- = (1/2)|a(c-b) + b(a-c) + c(b-a)|
- = (1/2)|ac - ab + ab - bc + bc - ac|
- = (1/2)|0| = 0
Since the area is always 0, the points are always collinear.
Hence proved.
Example 8: Finding Two Unknown Values
Problem: If the points (a, 0), (0, b), and (3, 2) are collinear, express the relationship between a and b.
Solution:
Area = 0 for collinearity:
- a(b - 2) + 0(2 - 0) + 3(0 - b) = 0
- ab - 2a + 0 - 3b = 0
- ab - 2a - 3b = 0
- ab = 2a + 3b
Dividing by ab: 1 = 2/b + 3/a, i.e., 3/a + 2/b = 1.
Answer: The relationship is 3/a + 2/b = 1.
Example 9: Collinearity of Division Points
Problem: Show that the midpoint of A(2, 4) and B(6, 8) is collinear with A and B.
Solution:
Step 1: Midpoint M = ((2+6)/2, (4+8)/2) = (4, 6)
Step 2: Using area method with A(2,4), M(4,6), B(6,8):
- Area = (1/2)|2(6-8) + 4(8-4) + 6(4-6)|
- = (1/2)|(-4) + 16 + (-12)|
- = (1/2)|0| = 0
Area = 0, so A, M, B are collinear.
This is expected since the midpoint always lies on the line segment joining the two points.
Example 10: Word Problem on Collinearity
Problem: Three friends live at locations represented by coordinates A(1, 2), B(4, 5), C(7, 8) on a city map. Determine whether they live along the same straight road.
Solution:
- AB = sqrt(9+9) = sqrt(18) = 3sqrt(2)
- BC = sqrt(9+9) = sqrt(18) = 3sqrt(2)
- AC = sqrt(36+36) = sqrt(72) = 6sqrt(2)
AB + BC = 3sqrt(2) + 3sqrt(2) = 6sqrt(2) = AC
Answer: Yes, they live along the same straight road.
Example 11: Collinearity of Parametric Points
Problem: Show that the points (2t, 5t-1), (3t+1, 7t), (t-1, 3t-2) are collinear for t = 1.
Solution:
Substituting t = 1:
- Point A = (2, 4), Point B = (4, 7), Point C = (0, 1)
Area method:
- = (1/2)|2(7-1) + 4(1-4) + 0(4-7)|
- = (1/2)|12 + (-12) + 0|
- = (1/2)|0| = 0
Since area = 0, the points are collinear when t = 1.
Example 12: Finding Multiple Values of k for Collinearity
Problem: For what values of k are the points (k, 2-2k), (1-k, 2k), and (-4-k, 6-2k) collinear?
Solution:
Using area method (set to 0):
- k(2k - (6-2k)) + (1-k)((6-2k) - (2-2k)) + (-4-k)((2-2k) - 2k) = 0
- k(4k - 6) + (1-k)(4) + (-4-k)(2-4k) = 0
- 4k^2 - 6k + 4 - 4k + (-8 + 16k - 2k + 4k^2) = 0
- 4k^2 - 6k + 4 - 4k - 8 + 16k - 2k + 4k^2 = 0
- 8k^2 + 4k - 4 = 0
- 2k^2 + k - 1 = 0
- (2k - 1)(k + 1) = 0
- k = 1/2 or k = -1
Answer: k = 1/2 or k = -1.
Example 13: Collinearity Using Section Formula
Problem: Show that the point P(3, 7) lies on the line joining A(1, 3) and B(5, 11).
Solution:
Method 1 (Area):
- = (1/2)|1(7-11) + 3(11-3) + 5(3-7)|
- = (1/2)|(-4) + 24 + (-20)|
- = (1/2)|0| = 0
Since area = 0, the points are collinear.
Method 2 (Section formula): If P divides AB in ratio k:1:
- 3 = (5k + 1)/(k + 1) → 3k + 3 = 5k + 1 → k = 1
- Check: 7 = (11(1) + 3)/(1 + 1) = 14/2 = 7. Verified.
P divides AB in ratio 1:1 (P is the midpoint of AB).
Real-World Applications
Applications of collinearity:
- Geometry proofs: Proving that three specific points lie on the same line is essential in many Euclidean geometry proofs.
- Navigation: Checking whether a ship or aircraft is on course between two waypoints — if its position is collinear with the waypoints, it is on the correct path.
- Surveying: Verifying that fence posts or boundary markers lie along a straight line.
- Computer graphics: Detecting degenerate triangles (zero area) in mesh generation.
- Physics: Checking if three force vectors act along the same line (concurrent or collinear forces).
- Data analysis: Three data points are perfectly linearly related if and only if they are collinear.
Key Points to Remember
- Collinear points lie on the same straight line.
- Distance method: Points A, B, C are collinear if AB + BC = AC (where AC is the longest distance).
- Area method: Points are collinear if x1(y2-y3) + x2(y3-y1) + x3(y1-y2) = 0.
- Slope method: Points are collinear if slope of AB = slope of BC.
- The area method is often the fastest for numerical problems.
- For finding unknown coordinates, set the area expression to zero and solve.
- The midpoint of two points is always collinear with those two points.
- Any point on a line segment is collinear with the endpoints.
- If three points are NOT collinear, they form a triangle with non-zero area.
- The distance method also confirms which point lies between the other two.
Practice Problems
- Check whether the points (3, 7), (6, 5), (15, -1) are collinear using the distance formula.
- Using the area method, determine if the points (-1, -1), (2, 3), (8, 11) are collinear.
- Find the value of p for which points (2, -1), (p, 1), (11, 4) are collinear.
- Prove that the points (a, 0), (0, b), (1, 1) are collinear if 1/a + 1/b = 1.
- Determine whether the points (cos A, cos B), (sin A, sin B), (0, 0) are collinear for A = 30 degrees, B = 60 degrees.
- The points (k, 3), (6, -2), (-3, 4) are collinear. Find k.
- Show that the points (2t, 5t), (3t, 7t), (5t, 11t) are collinear for all values of t.
- Three villages are located at coordinates (2, 3), (5, 6), (8, 9). Are they on the same road?
Frequently Asked Questions
Q1. What does collinear mean?
Collinear points are points that lie on the same straight line. Three or more points can be tested for collinearity.
Q2. How do you check collinearity using the distance formula?
Find all three distances AB, BC, AC. If the sum of the two shorter distances equals the longest distance (e.g., AB + BC = AC), the points are collinear.
Q3. How do you check collinearity using the area method?
Substitute the three points into the area formula. If the area of the triangle formed = 0, the points are collinear.
Q4. Which method is easier — distance or area?
The area method is usually easier because it avoids square roots and involves only multiplication and addition. The distance method requires computing square roots.
Q5. Can the slope method be used for collinearity?
Yes. If slope of AB = slope of BC, the points are collinear. However, this method fails if any pair has an undefined slope (vertical line). Use with caution.
Q6. What if two of the three points are the same?
If two points coincide, all three points are trivially collinear (any two points determine a line, and the coincident point lies on that line).
Q7. How do you find an unknown coordinate for collinearity?
Use the area method: set x1(y2-y3) + x2(y3-y1) + x3(y1-y2) = 0, substitute known values, and solve the resulting linear equation for the unknown.
Q8. Is collinearity a common board exam question?
Yes. Checking collinearity and finding unknown coordinates for collinearity are frequently asked in CBSE Class 10 board exams, worth 2-3 marks.
Q9. Can four points be collinear?
Yes. Four or more points can be collinear if they all lie on the same straight line. To check, verify collinearity for every combination of three points, or simply check that the slope between every consecutive pair is the same.
Q10. What is the geometric interpretation of collinearity?
Collinear points lie on a single straight line. The triangle they would form is degenerate — it has zero area and zero width. Visually, the three points can be connected by a single straight ruler without bending.
Related Topics
- Distance Formula
- Area of Triangle Using Coordinates
- Collinear and Non-Collinear Points
- Section Formula
- Introduction to Coordinate Geometry
- Cartesian Plane
- Plotting Points in Four Quadrants
- Abscissa and Ordinate
- Midpoint Formula
- Centroid of a Triangle
- External Division (Section Formula)
- Coordinate Geometry Word Problems
- Equation of a Line (Introduction)
- Quadrilateral Using Coordinate Geometry
