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Understanding Coordinate Geometry Concepts

May 25, 2025

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Lecture Notes: Geometry and Coordinate Geometry

Overview

This lecture focuses on solving problems related to coordinate geometry, specifically involving parallelograms, rectangles, and various geometric calculations using formulas. Key topics include distance and midpoint formulas, gradient calculations, proving perpendicularity, and understanding the properties of geometric shapes like parallelograms and rectangles.

Key Questions and Concepts

  1. Distance Formula

    • Used to calculate the length of the diagonal DB.
    • Formula: ( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
    • Example: Calculating length of DB using points: Point 2 (x2, y2) and Point 1 (x1, y1).
    • Result: Length of DB = 10.

  2. Midpoint Formula

    • Used to calculate the coordinates of the midpoint (M) of the diagonal DB.
    • Formula: ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) )
    • Result: Midpoint of DB = (0, -1).

  3. Gradient (Slope) Formula

    • Used to calculate the gradient of line AD.
    • Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
    • Result: Gradient of AD = 0.5.

  4. Proving Perpendicular Lines

    • Perpendicular lines have gradients that multiply to -1.
    • Prove AD is perpendicular to AB by calculating gradients:
      • Gradient of AB = -2.
      • Multiplication: ( 0.5 \times (-2) = -1 ).
    • Conclusion: AD and AB are perpendicular.

  5. Properties of Parallelogram and Rectangle

    • A rectangle is a parallelogram with one internal angle of 90°.
    • Proving Rectangle: If one angle is 90° in a parallelogram, all angles will be 90° due to properties of parallel lines and angles.

  6. Equation of a Line

    • Determine equation of KL given as ( y = mx + c ).
    • KL is parallel to AD, hence shares the same gradient (0.5).
    • Use point K to find y-intercept (c): ( y = \frac{1}{2}x - \frac{9}{4} ).

  7. Determining Coordinates of C

    • Using properties of parallelograms, calculate C using coordinate transformation:
    • From A to D, reduce x by 4 and y by 2; apply same to B for C.
    • Result: Coordinates of C = (0, -6).

Additional Notes

  • Diagonal Properties: Diagonals in a rectangle are equal.
  • Inspection Method: A method to determine points using the relationship and transformations between known points.

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