Length and midpoint
Length of a segment (Extended)
- A line segment is the straight piece between two points.
- Its length comes from Pythagoras on the horizontal and vertical gaps:
$$\text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
- Example: from $(1, 2)$ to $(4, 6)$ → $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
Practice
Find the length of the segment from (1, 2) to (4, 6).
√((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.
Midpoint (Extended)
- The midpoint is the average of the coordinates:
$$\left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$$
- Example: from $(1, 2)$ to $(4, 6)$ → $\left( \tfrac{5}{2}, \tfrac{8}{2} \right) = (2.5,\, 4)$.
Practice
The midpoint of (1, 2) and (4, 6) is (a, 4). What is a?
Average the x-values: (1 + 4)/2 = 2.5.
Practice
The midpoint of (2, 3) and (8, 9) is (5, b). What is b?
Average the y-values: (3 + 9)/2 = 6.
You've got it
Key idea
- length $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (Pythagoras)
- midpoint = average each coordinate
- $(1,2)$ to $(4,6)$: length $5$, midpoint $(2.5, 4)$