Lecture on Finding Midpoints

Introduction

• Objective: Learn how to find the midpoint between two points.
• Example Points: Point A (2,6) and Point B (4,10).

Formula for Midpoint

• Midpoint Formula:
  • X-coordinate: ( \frac{x_1 + x_2}{2} )
  • Y-coordinate: ( \frac{y_1 + y_2}{2} )

Calculating Midpoint

• Example 1:
  • Given Points: A (2,6) and B (4,10).
  • X-coordinate:
    • ( x_1 = 2, x_2 = 4 )
    • Calculation: ( \frac{2 + 4}{2} = 3 )
  • Y-coordinate:
    • ( y_1 = 6, y_2 = 10 )
    • Calculation: ( \frac{6 + 10}{2} = 8 )
  • Midpoint: (3, 8)

Practice Problem

• Given: A (3,5) and B (9,1).
  • X-coordinate:
    • Calculation: ( \frac{3 + 9}{2} = 6 )
  • Y-coordinate:
    • Calculation: ( \frac{5 + 1}{2} = 3 )
  • Midpoint: (6, 3)

Complex Example with Negative Numbers

• Given: A (-4, 2) and B (8, -6).
  • X-coordinate:
    • Calculation: ( \frac{-4 + 8}{2} = 2 )
  • Y-coordinate:
    • Calculation: ( \frac{2 - 6}{2} = -2 )
  • Midpoint: (2, -2)

Example with Fractions

• Given: A (1/3, 2) and B (5, -1/4).
  • X-coordinate:
    • Calculation: ( \frac{1/3 + 5}{2} )
    • Simplification: Multiply numerator and denominator by 3.
    • Final Value: ( \frac{16}{6} = \frac{8}{3} )
  • Y-coordinate:
    • Calculation: ( \frac{2 - 1/4}{2} )
    • Simplification: Multiply top and bottom by 4.
    • Final Value: ( \frac{7}{8} )
  • Midpoint: (8/3, 7/8)

Advanced Example with Multiple Fractions

• Given: A (1/5, -2/3) and B (4/3, 3/4).
  • X-coordinate:
    • Calculation: ( \frac{1/5 + 4/3}{2} )
    • Simplification: Multiply by 15.
    • Final Value: ( \frac{23}{30} )
  • Y-coordinate:
    • Calculation: ( \frac{-2/3 + 3/4}{2} )
    • Simplification: Multiply by 12.
    • Final Value: ( \frac{1}{24} )
  • Midpoint: (23/30, 1/24)

Conclusion

• Recap: The process of finding the midpoint remains consistent regardless of whether you're dealing with whole numbers, negatives, or fractions.
• Practice: Regularly practice the formula and calculations to improve speed and accuracy.