Coordinate Geometry Applications

📋 Course Outline

1. Distance Between Two Points
2. Collinearity Through Distances
3. Classifying Triangles
4. Midpoints and Unknown Coordinates
5. Circles and Coordinate Tests
6. Coordinate Geometry Applications

📖 1. Distance Between Two Points

📝 Essential Points

📐 Formula — The distance between points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

• For points I=(5,0) and N=(0,-6), the distance IN is $\sqrt{61}$ units.

💡 Memory Hook

Subtract coordinates, square, add, then take the square root

📖 2. Collinearity Through Distances

📝 Essential Points

📌 Three points are collinear when the sum of two consecutive distances equals the distance between the two outer points.

• For A=(-2,0), B=(4,8) and C=(4,-12), AB=10 units, BC=√85 units and AC=√202 units; since AB+BC is not equal to AC, the points are not collinear.

💡 Memory Hook

Equal distance sums imply collinearity; unequal sums imply non-collinearity

📖 3. Classifying Triangles

📝 Essential Points

• For A=(0,0), B=(6,8) and R=(-3,-4), the distances AB and AR are both 10 units, supporting an isosceles triangle.

• For Q=(0,0), A=(-4,0) and B=(0,4), QA=QB=4 units and the triangle is right-angled isosceles.