Class 10 / Maths
Class 10 Maths Coordinate Geometry Notes
Revision notes on distance formula, section formula, midpoint formula, and area of a triangle.
Topic introduction
Coordinate Geometry converts geometry questions into calculations using points on the plane. Most questions need careful substitution in distance, midpoint, section, or area formulas.
Key points
- The distance formula finds the length between two points on the coordinate plane.
- The section formula finds a point that divides a line segment in a given ratio.
- The midpoint formula is a special case of the section formula when the ratio is 1:1.
- The area formula helps check whether three points are collinear.
Definitions
Coordinate plane: A plane formed by the x-axis and y-axis where points are written as ordered pairs.
Ordered pair: A point written as (x, y), where x is the horizontal coordinate and y is the vertical coordinate.
Collinear points: Three or more points that lie on the same straight line.
Section formula: A formula used to find a point that divides a line segment in a given ratio.
Formulas
- Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
- Internal section point = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
- Area of triangle = 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
Examples
- Distance between (3, 4) and (7, 1) is sqrt((4)^2 + (-3)^2) = 5.
- Midpoint of (6, -2) and (-4, 8) is (1, 3).
- If triangle area is zero, the three points are collinear.
Distance and midpoint
For points (x1, y1) and (x2, y2), distance is the square root of (x2 - x1)^2 + (y2 - y1)^2.
The midpoint is ((x1 + x2)/2, (y1 + y2)/2). Use it when a line segment is divided into two equal parts.
Section formula and area
If a point divides the segment joining two points in the ratio m:n internally, apply the section formula carefully with matching x and y coordinates.
The area of a triangle with three coordinate points is useful for geometry questions and for proving collinearity when the area is zero.
Common mistakes
- Subtracting x-coordinate from y-coordinate in the distance formula.
- Forgetting the square root in distance questions.
- Reversing m and n in the section formula.
- Ignoring the absolute value in area of triangle.
- Writing points as y, x instead of x, y.
Quick practice
- Find the distance between (2, 3) and (6, 6).
- Find the midpoint of (-4, 7) and (8, 1).
- Find the point dividing (2, 3) and (8, 9) in the ratio 1:2.
- Check whether (1, 2), (3, 4), and (5, 6) are collinear.
Related practice
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