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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 35
Chapter 3, Problem 35

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)

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1
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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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 M = ((x1 + x2)/2, (y1 + y2)/2). This formula is essential to relate the midpoint and endpoints.
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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.
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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.
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Graphs and Coordinates - Example
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