1.3 Midpoint and Endpoint Instructions.webm: McCarthy - Geometry

Midpoint and Endpoint Concepts

A line segment is a section of a line bounded by two distinct endpoints. The midpoint is the exact center point of a line segment, dividing it into two equal and congruent parts.

If $M$ is the midpoint of segment $AB$, then segment $AM$ is congruent to segment $MB$, represented as $AM = MB$.

Calculating the Midpoint of a Line Segment

To find the midpoint on a one-dimensional ($1D$) number line between two points $x_1$ and $x_2$, average the two values:

$M = \frac{x_1 + x_2}{2}$

To find the midpoint $M(x_m, y_m)$ in a two-dimensional ($2D$) coordinate plane between endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$, calculate the averages of the x-coordinates and y-coordinates independently:

$x_m = \frac{x_1 + x_2}{2}$

$y_m = \frac{y_1 + y_2}{2}$

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

For example, given endpoints $A(2, 4)$ and $B(8, 10)$, the midpoint $M$ is calculated as:

$x_m = \frac{2 + 8}{2} = 5$

$y_m = \frac{4 + 10}{2} = 7$

$M = (5, 7)$

Finding an Endpoint Given Midpoint and One Endpoint

When given one endpoint $A(x_1, y_1)$ and the midpoint $M(x_m, y_m)$, the missing endpoint $B(x_2, y_2)$ is found by isolating $x_2$ and $y_2$ in the midpoint equations:

$x_2 = 2x_m - x_1$

$y_2 = 2y_m - y_1$

For example, given endpoint $A(1, 3)$ and midpoint $M(4, 7)$, find missing endpoint $B(x_2, y_2)$:

$x_2 = 2 \times 4 - 1 = 7$

$y_2 = 2 \times 7 - 3 = 11$

$B = (7, 11)$

Practice and Interactive Resources

Key resources for learning and practice include:

1. 1.3 Midpoint and Endpoint Instructions.webm

2. Desmos: Midpoint Exploration

3. 1.3 Practice WS (Midpoints & Endpoints)