Lecture Notes: Calculating Distance and Midpoint between Two Points

Introduction

• Objective: Calculate the distance and coordinate of the midpoint of a line segment between two points.
• Example Points: (-2, -4) and (1, 5)

Plotting Points and Line Segment

• Plot points on a coordinate plane:
  • Point A: (-2, -4)
  • Point B: (1, 5)
• Draw a line segment connecting the points.

Finding Distance between Two Points

• Concept: Use the Pythagorean Theorem to find the hypotenuse, which represents the distance.
• Steps:
  • Draw horizontal and vertical lines to form a right triangle with the line segment as the hypotenuse.
  • Calculate differences (delta) for x and y coordinates:
    • ( \Delta x = x_2 - x_1 = 1 - (-2) = 3 )
    • ( \Delta y = y_2 - y_1 = 5 - (-4) = 9 )
  • Apply the Pythagorean Theorem:
    • ( d^2 = (\Delta x)^2 + (\Delta y)^2 )
    • ( d^2 = 3^2 + 9^2 = 9 + 81 = 90 )
    • ( d = \sqrt{90} = 3\sqrt{10} )

Distance Formula

• Formula: Derived from the Pythagorean Theorem.
  • ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
• Example Calculation:
  • Given points: (-2, -4) and (1, 5)
  • ( d = \sqrt{(1 - (-2))^2 + (5 - (-4))^2} )
  • ( d = \sqrt{3^2 + 9^2} = \sqrt{90} = 3\sqrt{10} )

Conclusion

• Summary: The distance formula simplifies finding the distance between two points without needing to plot or draw triangles.
• Usage: Use the formula for any given pair of points to quickly determine the distance.