Schooling · Class 10 · Mathematics · Coordinate Geometry · Chapter on a page
CBSE Class 10 · Mathematics · NCERT Mathematics
Coordinate Geometry: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Every point in a plane is uniquely…
- The distance formula gives the…
- The section formula finds the exact…
- The midpoint is a special case of…
- The centroid of a triangle is the…
- Coordinates let you verify geometric…
- Formulas and facts
- Distance formula
- Section formula (internal division)
- Midpoint formula
- Centroid of a triangle
- Collinearity check
- Watch out for
- Forgetting to square before adding…
- Swapping m and n in the section…
- Taking the square root of each…
- Ignoring negative signs when…
- Using section formula when midpoint…
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.
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 formulaDistance 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, thenR = ((m × x₂ + n × x₁)/(m + n), (m × y₂ + n × y₁)/(m + n))Meaning: Weighted average of coordinates, with weights from the ratio.
- Midpoint formulaMidpoint of PQ = ((x₁ + x₂)/2, (y₁ + y₂)/2)Meaning: Simply average the x-coordinates and average the y-coordinates.
- Centroid of a triangleCentroid 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 checkThree 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.
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.