Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b).
midpoint $(12,6)$, endpoint $(19,16)$

Verified step by step guidance

1. Recall the midpoint formula for a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\): the midpoint \((M_x, M_y)\) is given by \(M_x = \frac{x_1 + x_2}{2}\) and \(M_y = \frac{y_1 + y_2}{2}\).
2. Identify the known values: the midpoint is \((12, 6)\), and one endpoint is \((19, 16)\). Let the unknown endpoint be \((x, y)\).
3. Set up equations using the midpoint formula: \(12 = \frac{19 + x}{2}\) and \(6 = \frac{16 + y}{2}\).
4. Solve each equation for the unknown coordinate: multiply both sides by 2 to eliminate the denominator, then isolate \(x\) and \(y\).
5. Write the coordinates of the other endpoint as \((x, y)\) using the values found from the previous step.

Key Concepts

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 for relating the midpoint to the 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 you to determine the missing endpoint.

Coordinate Geometry

Coordinate geometry involves using algebraic methods to solve geometric problems on the coordinate plane. Understanding how points, lines, and segments relate through their coordinates helps in visualizing and solving problems like finding endpoints or midpoints.