Definition
Coordinate Geometry (also called Cartesian Geometry) is the branch of mathematics in which the position of points on a plane is described using ordered pairs called coordinates. The plane is divided by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).
Important Formulas
1. Distance Formula
The distance between two points ( A(x_1, y_1) ) and ( B(x_2, y_2) ):
[ AB = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} ]
2. Section Formula
If a point ( P(x, y) ) divides the line segment joining ( A(x_1, y_1) ) and ( B(x_2, y_2) ) in the ratio ( m:n ):
[ x = \frac{mx_2 + nx_1}{m + n} ] [ y = \frac{my_2 + ny_1}{m + n} ]
3. Midpoint Formula
The midpoint ( M ) of the line segment joining ( A(x_1, y_1) ) and ( B(x_2, y_2) ):
[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]
4. Area of a Triangle
The area of a triangle with vertices ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ):
[ \text{Area} = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right| ]
Example Problems
1. Find the distance between the points ( (2, 3) ) and ( (7, 11) ).
[ AB = \sqrt{(7 – 2)^2 + (11 – 3)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43 ]
2. Find the coordinates of the point dividing the line joining ( (1, 2) ) and ( (5, 10) ) in the ratio 1:3.
[ x = \frac{1 \times 5 + 3 \times 1}{1 + 3} = \frac{5 + 3}{4} = 2 ] [ y = \frac{1 \times 10 + 3 \times 2}{1 + 3} = \frac{10 + 6}{4} = 4 ] So, the point is ( (2, 4) ).
3. Find the area of the triangle with vertices ( (1, 1) ), ( (4, 5) ), and ( (7, 2) ).
[ \text{Area} = \frac{1}{2} |1(5-2) + 4(2-1) + 7(1-5)| ] [ = \frac{1}{2} |1 \times 3 + 4 \times 1 + 7 \times (-4)| ] [ = \frac{1}{2} |3 + 4 – 28| = \frac{1}{2} |-21| = 10.5 ]
Summary Table
| Concept | Formula | Description |
|---|---|---|
| Distance | ( \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} ) | Distance between two points |
| Section | ( \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) ) | Point dividing a segment |
| Midpoint | ( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ) | Middle point of a segment |
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