Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b).
midpoint $open paren 5 comma 8 close paren$ , endpoint $open paren 13 comma 10 close paren$

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Recall the midpoint formula: if the midpoint is $M(x_m, y_m)$ and the endpoints are $A(x_1, y_1)$ and $B(x_2, y_2)$, then the midpoint coordinates are given by $x_m = \frac{x_1 + x_2}{2}$ and $y_m = \frac{y_1 + y_2}{2}$.

Identify the known values: the midpoint $M$ is $(5, 8)$, and one endpoint $A$ is $(13, 10)$. We need to find the other endpoint $B(x_2, y_2)$.

Set up equations using the midpoint formula for each coordinate: $5 = \frac{13 + x_2}{2}$ and $8 = \frac{10 + y_2}{2}$.

Solve each equation for the unknown coordinate: multiply both sides by 2 to get $10 = 13 + x_2$ and $16 = 10 + y_2$, then isolate $x_2$ and $y_2$ by subtracting the known values.

Write the coordinates of the other endpoint $B$ as $(x_2, y_2)$ after solving the equations.

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주요 개념

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Midpoint Formula

The midpoint formula calculates the point exactly halfway between two endpoints of a line segment. It is found by averaging the x-coordinates and the y-coordinates of the endpoints separately: Midpoint M = ((x1 + x2)/2, (y1 + y2)/2). This formula is essential to relate the midpoint and endpoints.

Solving for an Unknown Endpoint

Given the midpoint and one endpoint, you can find the other endpoint by rearranging the midpoint formula. Multiply the midpoint coordinates by 2, then subtract the known endpoint coordinates: (x2, y2) = (2 * midpoint_x - x1, 2 * midpoint_y - y1). This allows determination of the missing endpoint.

Coordinate Geometry Basics

Understanding how points are represented in the coordinate plane and how to manipulate their coordinates is fundamental. This includes recognizing ordered pairs (x, y) and performing arithmetic operations on these values to solve geometric problems involving lines and segments.

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (5,8)(5, 8), endpoint (13,10)(13, 10)