Algebra Lecture Notes: Distance and Midpoint Formulas, Pythagorean Theorem

Overview

• Focus on distance formula, midpoint formula, and Pythagorean theorem.
• These concepts are fundamental in coordinate geometry and have applications in advanced sciences like physics.

Pythagorean Theorem

• Definition: In a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
  • Formula: ( c^2 = a^2 + b^2 )
• Only holds in flat (Euclidean) space.
• Importance in advanced science, e.g., measuring curvature in space-time.

Example

• Right Triangle with sides 3, 4, 5
  • Check: ( 5^2 = 3^2 + 4^2 )
  • ( 25 = 9 + 16 )
  • Confirms a right triangle.
• Incorrect Example with sides 1, 9, 11 (not right triangle).

Distance Formula

• Derived from Pythagorean Theorem: Measures distance between two points in a plane.
  • Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
• Represents the hypotenuse of a right triangle formed by the two points and their projections on the axes.

Example

• Points: ((-1, 2), (3, 4))
  • Calculation: ( d = \sqrt{(3 + 1)^2 + (4 - 2)^2} )
  • Results in: ( d = 2 \sqrt{5} )

Midpoint Formula

• Purpose: To find the midpoint between two points on a plane.
  • Formula: Midpoint ( M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) )
• Essentially averages the x and y coordinates.

Example

• Points: ((4, -6), (-3, 2))
  • Midpoint Calculation: ( M = (\frac{4 - 3}{2}, \frac{-6 + 2}{2}) = (\frac{1}{2}, -2) )

Applications and Importance

• Understanding these formulas is crucial for higher-level math and sciences like physics and engineering.
• The Pythagorean theorem has parallels in theories such as Einstein's General Theory of Relativity.

Conclusion

• These foundational concepts bridge algebra and geometry, providing tools for measurement and analysis in mathematics and science.
• Further exercises will involve more complex applications of these formulas.