Coordinate Geometry Revision Guide

Key Concepts

• Gradient of a Line Segment
  • Formula: (m = \frac{y_2 - y_1}{x_2 - x_1})
  • Alternative method: (m = \tan(\theta)) where (\theta) is the angle with the positive x-axis.

Example 1: Finding Equation of Line AC

• Given points A, B, C (collinear) and (\tan(\theta) = -2)
• Gradient Calculation: (m = -2)
• Equation of Line: (y - y_1 = m(x - x_1))
• Result: (y = -2x + 4)

Example 2: Finding Equation with Angle 45 degrees

• Given (\alpha = 45^{\circ}), find (\theta = 135^{\circ})
• Gradient Calculation: (m = \tan(135^{\circ}) = -1)
• Result: (y = -x + 3)

Example 3: Gradients of Parallel Lines

• Two lines: (y = m_1x + c_1) and (y = m_2x + c_2)
• If parallel: (m_1 = m_2)
• Example: Line parallel to (x + 2y = 6)
  • Rearrange: (y = -\frac{1}{2}x + 3) (Gradient = -1/2)
  • Find coordinates where this line meets x-axis, result: (11, 0).

Example 4: Collinearity of Points

• Check if three points P, Q, R are collinear by equating gradients of PQ and QR.

Midpoint Formula

• For points A and B, the midpoint is: (M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right))
• Useful for finding fourth vertex in shapes like parallelograms and rectangles.

Perpendicular Lines

• If two lines are perpendicular, then (m_1 \cdot m_2 = -1)
• Finding Equation of Perpendicular Bisector: Use midpoint and negative reciprocal of gradient.

Distance Formula

• To find distance between two points (A(x_1, y_1)) and (B(x_2, y_2)):
  [d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]

Area of Shapes using Shoelace Method

• For a polygon with vertices ( (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) ): [ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n}(x_iy_{i+1} - y_ix_{i+1}) \right|]
• Important to arrange points in anti-clockwise order._

Properties of Special Quadrilaterals

• Rectangle: Four right angles, opposite sides equal.
• Square: Four equal sides, four right angles.
• Parallelogram: Opposite sides equal and parallel.
• Rhombus: All sides equal, diagonals intersect at 90 degrees.
• Trapezium: One pair of parallel sides.
• Kite: Two pairs of adjacent equal sides, diagonals intersect at 90 degrees.

Example Problems

1. Trapezium Problem: Find coordinates given midpoint and parallel lines.
2. Quadrilateral ABCD: Determine area using the shoelace method and check properties.
3. Triangle Problem: Use area and perpendicularity to find coordinates.

Conclusion

• Master these concepts for effective problem-solving in Coordinate Geometry. Happy studying!