Understanding 2D Geometry Concepts and Formulas

Sep 17, 2024

Lecture Notes on 2D Geometry Prerequisites

Introduction

  • Welcome by Mekiran sir on Vedanta Telugu Jai channel.
  • Focus on 2D geometry prerequisites, including concepts, tricks, and shortcuts.
  • Classes will be held in one shot; if chapters are large, two classes will be conducted.

Key Topics Covered

Distance Between Two Points

  • Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
  • Order of points does not affect the result due to squaring.

Section Formula

  • Internal Section Formula:
    [ C = \left( \frac{Mx_2 + Nx_1}{M + N}, \frac{My_2 + Ny_1}{M + N} \right) ]
  • External Section Formula:
    [ C = \left( \frac{Mx_2 - Nx_1}{M - N}, \frac{My_2 - Ny_1}{M - N} \right) ]
  • Midpoint Formula:
    [ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Finding Ratio

  • To find the ratio dividing two points, use cross multiplication.

Harmonic Conjugates

  • Points P and Q are harmonic conjugates if they divide the segment internally and externally in the same ratio.

Points of Trisection

  • Points dividing a segment into three equal parts.
  • Ratio examples: 1:2 and 2:1.

Collinearity

  • Collinear Points: Points are collinear if:
    1. The sum of lengths of any two sides equals the third side.
    2. The area of the triangle formed by the points is zero.
    3. Slopes of pairs of points are equal.

Area of Triangle

  • Area using Determinants:
    [ Area = \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1)| ]

Centroid

  • Centroid formula:
    [ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]
  • Divides medians in a 2:1 ratio.

Incenter

  • The intersection of angle bisectors.
  • Incenter formula:
    [ I = \left( \frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} \right) ]

Circumcenter and Orthocenter

  • Circumcenter is the intersection of the perpendicular bisectors.
  • Orthocenter is where the altitudes meet.

Quadrilaterals

  • Various types: Parallelogram, Rhombus, Rectangle, Square.
  • Properties of diagonals and sides differentiate types.

Locus

  • Definition: The path traced by a point moving according to certain conditions.
  • Important to eliminate variables (x, y) in the final locus equation.

Example Problems

  1. Finding the centroid of a triangle with vertices at given points.

    • Use centroid formula with the found points to calculate.

  2. Finding diagonals and other properties in a parallelogram.

    • Understand that diagonals bisect in parallelograms.

  3. Equations of angle bisectors using the section formula.

    • Apply ratios to find angle bisectors.

Conclusion

  • Encourage students to engage and practice more problems for mastery.
  • Reminder to register for MVSAT and join new Telegram channel for updates and resources.

Note: The formulas and examples should be practiced to gain a better understanding of 2D geometry concepts.