Study sheet: 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.

💡 Memory Hook

Equal sides identify isosceles triangles, while equal perpendicular sides identify right-angled isosceles triangles

📖 4. Midpoints and Unknown Coordinates

📝 Essential Points

📐 Formula — The midpoint of points $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

• 🔄 Finding an unknown midpoint coordinate involves these steps:

  1. Substitute the known endpoint coordinates into the midpoint formula
  2. Form the corresponding coordinate equation
  3. Solve for the unknown coordinate

• The midpoint of S=(-3,0) and T=(3,0) is M=(0,0).

💡 Memory Hook

The midpoint sits halfway between two endpoints, balancing their coordinates

📖 5. Circles and Coordinate Tests

📝 Essential Points

📌 For a circle centered at the origin, a point is inside the circle when its distance from the origin is less than the radius and outside when its distance is greater than the radius.

• For D=(5,6), the distance from the origin is √61, which is less than √65, so D lies inside the circle of radius √65.

💡 Memory Hook

A point is inside when its distance is smaller than the radius, and outside when it is larger

📖 6. Coordinate Geometry Applications

📝 Essential Points

• The distance between screen points (100,150) and (250,230) is 170 pixels.

📊 Synthesis Tables

Coordinate Geometry Tests

Test	Condition	Conclusion
Collinearity	Sum of two distances equals the third	Points lie on one straight line
Isosceles triangle	Two side lengths are equal	Triangle is isosceles
Circle position	Distance less than radius	Point is inside
Circle position	Distance greater than radius	Point is outside