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. Let the coordinates of the unknown endpoint be \((x, y)\).
2. Use the midpoint formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (12, 6) \).
3. Substitute the known endpoint \((19, 16)\) and midpoint \((12, 6)\) into the formula: \( \left( \frac{19 + x}{2}, \frac{16 + y}{2} \right) = (12, 6) \).
4. Set up the equations: \( \frac{19 + x}{2} = 12 \) and \( \frac{16 + y}{2} = 6 \).
5. Solve each equation for \(x\) and \(y\) to find the coordinates of the other endpoint.

Key Concepts

Midpoint Formula

The midpoint formula is used to find the midpoint of a line segment given its endpoints. It is calculated as M = ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints. This concept is essential for determining the coordinates of the other endpoint when one endpoint and the midpoint are known.

Coordinate Geometry

Coordinate geometry involves the study of geometric figures using a coordinate system. It allows for the representation of points, lines, and shapes in a two-dimensional space using ordered pairs (x, y). Understanding how to manipulate these coordinates is crucial for solving problems related to line segments and their properties.

Solving for Unknowns

In algebra, solving for unknowns involves using equations to find missing values. When given the midpoint and one endpoint, you can set up equations based on the midpoint formula to solve for the coordinates of the unknown endpoint. This process is fundamental in algebra and helps in understanding relationships between different points in a coordinate plane.