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Coordinate Geometry

Master the intersection of Algebra and Geometry: Lines, Slopes, and Shapes on the Cartesian Plane.

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1. The Cartesian Plane & Distance

Quadrants:

Q1 (+, +): Top Right
Q2 (-, +): Top Left
Q3 (-, -): Bottom Left
Q4 (+, -): Bottom Right

Distance Formula

Distance between $A(x_1, y_1)$ and $B(x_2, y_2)$ :

D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

2. Section Formula & Centroid

Internal Division (Ratio m:n)

$P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$

Midpoint (Ratio 1:1)

$M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Centroid of Triangle

Intersection of Medians:

$G(x, y) = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right)$

3. Area of Triangle (Shoelace Method)

For vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ :

Area = $\frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |$

Note: If Area = 0, points are Collinear (Lie on the same line).

4. Slope & Equation of Line

Slope ( $m$ ): $m = \tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$
Equation (Slope-Point): $y - y_1 = m(x - x_1)$
Standard Form: $ax + by + c = 0$ $\to$ Slope $= -a/b$

Parallel Lines

m_1 = m_2

Perpendicular

m_1 \times m_2 = -1

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Coordinate Geometry

1. The Cartesian Plane & Distance

Distance Formula

2. Section Formula & Centroid

Internal Division (Ratio m:n)

Midpoint (Ratio 1:1)

Centroid of Triangle

3. Area of Triangle (Shoelace Method)

4. Slope & Equation of Line

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