What this chapter is about
Coordinate geometry connects algebra with geometry by using numbers to describe positions and shapes on a plane. You learn to locate points using ordered pairs, find distances between points, and determine where a point lies that divides a line segment in a given ratio.
This chapter builds on the coordinate system you met in earlier classes and prepares you for more advanced work in Class 11 and 12. The tools here let you solve geometric problems through calculation rather than construction, which is often more precise and always verifiable.
After studying this chapter, you should be able to calculate the distance between any two points, find the coordinates of a point dividing a segment internally, and determine the centroid of a triangle using coordinates alone.
Key ideas
- Every point in a plane is uniquely located by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
- The distance formula gives the length of the segment joining two points without measuring it physically.
- The section formula finds the exact coordinates of a point that divides a segment in a given ratio m : n.
- The midpoint is a special case of the section formula where the ratio is 1 : 1.
- The centroid of a triangle is the point where all three medians meet, and its coordinates are the averages of the vertices' coordinates.
- Coordinates let you verify geometric properties (collinearity, type of triangle, type of quadrilateral) through calculation.
Formulas and facts to remember
Distance formula Distance between P(x₁, y₁) and Q(x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²] Meaning: Add the squares of horizontal and vertical differences, then take the square root.
Section formula (internal division) If point R divides segment PQ in ratio m : n, then R = ((m × x₂ + n × x₁)/(m + n), (m × y₂ + n × y₁)/(m + n)) Meaning: Weighted average of coordinates, with weights from the ratio.
Midpoint formula Midpoint of PQ = ((x₁ + x₂)/2, (y₁ + y₂)/2) Meaning: Simply average the x-coordinates and average the y-coordinates.
Centroid of a triangle Centroid of triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) Meaning: Each coordinate of the centroid is the mean of the three vertices' coordinates.
Collinearity check Three points are collinear if the area of the triangle they form equals zero, or equivalently if the sum of any two distances equals the third.
Worked examples
Example 1: Finding distance A mobile tower is at point A(3, 4) and a school is at point B(−1, 1). Find the distance between them in units.
Solution: Distance = √[(−1 − 3)² + (1 − 4)²] = √[(−4)² + (−3)²] = √[16 + 9] = √25 = 5 units
Example 2: Section formula Point P divides the segment joining A(2, −3) and B(7, 7) in the ratio 2 : 3. Find the coordinates of P.
Solution: x-coordinate of P = (2 × 7 + 3 × 2)/(2 + 3) = (14 + 6)/5 = 20/5 = 4 y-coordinate of P = (2 × 7 + 3 × (−3))/(2 + 3) = (14 − 9)/5 = 5/5 = 1 Therefore, P = (4, 1)
Example 3: Finding centroid A triangular park has vertices at R(0, 0), S(6, 0) and T(3, 9). Where should a water fountain be placed so it is equidistant from all three medians?
Solution: The fountain should be at the centroid. x-coordinate = (0 + 6 + 3)/3 = 9/3 = 3 y-coordinate = (0 + 0 + 9)/3 = 9/3 = 3 Centroid = (3, 3) The fountain should be placed at coordinates (3, 3).
Common mistakes
Forgetting to square before adding inside the distance formula → Always compute (x₂ − x₁)² and (y₂ − y₁)² separately first.
Swapping m and n in the section formula → Remember m is the part nearer to the first point, so the second point's coordinate is multiplied by m.
Taking the square root of each squared term separately → The root applies to the entire sum, not to individual parts.
Ignoring negative signs when subtracting coordinates → Subtracting a negative number gives addition; track signs carefully.
Using section formula when midpoint formula is simpler → When ratio is 1 : 1, just average the coordinates directly.
Quick revision
- Distance = √[(difference of x)² + (difference of y)²].
- Section formula weights coordinates by the opposite part of the ratio.
- Midpoint is the average of both coordinates.
- Centroid coordinates are means of the three vertices' coordinates.
- Collinear points lie on a single line; check using distance or area method.
- Always simplify square roots fully in final answers.