Length and Midpoint
3.4 Length and Midpoint – Cheat Sheet
1. Length of a Line Segment
- Formula (distance between two points (x₁, y₁) and (x₂, y₂)):
Length = √[(x₂ − x₁)² + (y₂ − y₁)²] - Derived from Pythagoras’ theorem.
Example:
Points A(2, 3) and B(7, 11):
Length = √[(7 − 2)² + (11 − 3)²] = √[(5)² + (8)²] = √(25 + 64) = √89
2. Midpoint of a Line Segment
- Formula:
Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Example:
Points A(2, 3) and B(7, 11):
Midpoint = ( (2 + 7)/2 , (3 + 11)/2 ) = (9/2 , 14/2) = (4.5 , 7)
3. Quick Reference Table
| Points A(x₁, y₁), B(x₂, y₂) | Length | Midpoint |
|---|---|---|
| (0, 0), (3, 4) | √[(3 − 0)² + (4 − 0)²] = √(9 + 16) = 5 | (1.5, 2) |
| (−2, 5), (4, −1) | √[(4 − (−2))² + (−1 − 5)²] = √[(6)² + (−6)²] = √72 = 6√2 | (1, 2) |
| (3, 8), (6, 12) | √[(6 − 3)² + (12 − 8)²] = √(9 + 16) = 5 | (4.5, 10) |