Midpoint and distance

Finding the midpoint of an interval, the length of an interval using the distance formula, and finding an endpoint from a midpoint.

Midpoint

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Drag A and B. The midpoint averages the coordinates; the distance comes from Pythagoras.

Example 1●

Find the midpoint of $(-3, 4)$ and $(5, -2)$.

$\left(\dfrac{-3 + 5}{2}, \dfrac{4 + (-2)}{2}\right) = (1, 1)$

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Example 2●●

Find the distance between $(-1, -2)$ and $(5, 6)$.

$d = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$

Example 3●●●

$M(4, 1)$ is the midpoint of $A(2, -3)$ and $B$. Find $B$.

$\dfrac{2 + x}{2} = 4$, so $x = 6$. $\dfrac{-3 + y}{2} = 1$, so $y = 5$.

$B = (6, 5)$

Flashcards

Midpoint

Average the $x$'s and the $y$'s.

Distance formula comes from

Pythagoras' theorem

Distance $(0,0)$ to $(3,4)$

$5$

Practice

Midpoint of $(-3, 4)$ and $(5, -2)$?
Distance between $(-1, -2)$ and $(5, 6)$?
$M(4, 1)$ is the midpoint of $A(2, -3)$ and $B$. Find $B$.
Midpoint of $(2, 3)$ and $(8, 9)$?
Distance between $(1, 2)$ and $(4, 6)$?
Exact distance from $(0, 0)$ to $(2, 3)$? Type a surd like sqrt(13).
Midpoint of $(-4, 1)$ and $(4, -4)$?
Distance between $(-2, 3)$ and $(10, -2)$?