Lecture Notes - Distance Formula in Coordinate Geometry

Introduction

• Topic: Distance formula in coordinate geometry
• Focus: How to calculate distance between two points using coordinates
• Practical applications in solving geometric problems

Key Concepts

Distance Formula

• Definition: The distance formula calculates the distance between two points in a coordinate plane.
• Formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
• Components:
  • \(x_1, y_1\): Coordinates of the first point
  • \(x_2, y_2\): Coordinates of the second point

Application Example 1

• Given Points: (0, 0) and (x, y)
• Calculation: [ d = \sqrt{x^2 + y^2} ]
• Explanation: Difference in x-coordinates and y-coordinates squared and summed, then take the square root

Application Example 2: Calculating Distance for Specific Points

• Example points: Points between (5, 3) and another point
  • Calculate using: [ d = \sqrt{(5 - 0)^2 + (3 - 0)^2} ]
  • Simplify to: [ d = \sqrt{25 + 9} = \sqrt{34} ]

Further Examples and Exercises

Example: Finding the Third Vertex in a Given Triangle

• Given Points: (-5, 3) and (5, 3)
• Determine the distance using the same formula
• Objective: Find coordinates of the third vertex, using conditions such as equal sides
• Equation Setup:
  • For triangle sides AC and BC to be equal
  • Equal distances imply same expressions for distance formula

Midsection Explanation

• Midpoint Formula: Useful for dividing line segment in a given ratio
• Formula: [ , M(x, y) = \left( , \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ]

Practice Problems

• Practice using given sets of points:
  • Example: (8, 2) and (1, -3)
  • Apply distance formula step-by-step

Special Triangles

• Isosceles Triangle: Two sides are equal
• Right-Angled Triangle: Pythagoras theorem applicable
  • Equations:
    • Sum of squares of two sides = Square of hypotenuse
    • Important for proving right angles

Conclusion

• Recap of distance formula and its importance in coordinate geometry
• Importance of practicing different types of coordinate problems
• Announcement for next class details (7:30 PM)

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