Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 38

Chapter 3, Problem 38

Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example 5(b). midpoint (-9, 8), endpoint (-16, 9)

Verified step by step guidance

Recall the midpoint formula for a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ \left( \frac{ x_1 + x_2 }{2}, \frac{ y_1 + y_2 }{2} \right) = (x_m, y_m) \] where \( (x_m, y_m) \) is the midpoint.

Identify the known values from the problem: the midpoint \( (x_m, y_m) = (-9, 8) \) and one endpoint \( (x_1, y_1) = (-16, 9) \). The other endpoint \( (x_2, y_2) \) is unknown.

Set up two equations using the midpoint formula by equating the midpoint coordinates to the average of the endpoints' coordinates: \[ \frac{ -16 + x_2 }{2} = -9 \] \[ \frac{ 9 + y_2 }{2} = 8 \]

Solve each equation for the unknown coordinate: For the \(x\)-coordinate: \[ -16 + x_2 = 2 \times (-9) \] \[ x_2 = 2 \times (-9) + 16 \] For the \(y\)-coordinate: \[ 9 + y_2 = 2 \times 8 \] \[ y_2 = 2 \times 8 - 9 \]

Write the coordinates of the other endpoint as \( (x_2, y_2) \) using the expressions found in the previous step.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 = ((x1 + x2)/2, (y1 + y2)/2). This formula is essential for relating endpoints to their midpoint.

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 = 2 * midpoint_x - x1, y2 = 2 * midpoint_y - y1. This method helps find 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 allows for precise calculations like finding midpoints and endpoints, which is fundamental in this problem.

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