Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the length of the line segment joining the midpoints of the segments \(A B\) and \(C D\), where \(A=(1,3), B=(2,6)\), \(C=(4,7)\), and \(D=(3,4)\).

Step-by-Step Solution

Answer

The length is \(\sqrt{5}\).

Step 1: Find the Midpoint of Segment AB

To find the midpoint of segment AB, use the midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). For points \(A(1,3)\) and \(B(2,6)\), the midpoint is \( \left( \frac{1+2}{2}, \frac{3+6}{2} \right) = \left( \frac{3}{2}, \frac{9}{2} \right) \).

Step 2: Find the Midpoint of Segment CD

For segment CD with points \(C(4,7)\) and \(D(3,4)\), use the midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substituting gives \( \left( \frac{4+3}{2}, \frac{7+4}{2} \right) = \left( \frac{7}{2}, \frac{11}{2} \right) \).

Step 3: Find the Distance Between the Midpoints

With midpoints \((\frac{3}{2}, \frac{9}{2})\) and \((\frac{7}{2}, \frac{11}{2})\), use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the values gives: \( d = \sqrt{\left(\frac{7}{2} - \frac{3}{2}\right)^2 + \left(\frac{11}{2} - \frac{9}{2}\right)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \).

Key Concepts

Midpoint Formula

The midpoint formula is a handy tool in coordinate geometry to determine the point that lies exactly halfway between two given points. This is particularly useful when analyzing line segments. The midpoint formula requires the coordinates of the two endpoints of the line segment.When given two endpoints, say \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the formula is expressed as:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula averages the \( x \) and \( y \) coordinates of the two endpoints, yielding the coordinates of the midpoint \( M(x, y) \). Applying this method to segments helps find the center between two points quickly and easily. Remember that using simple arithmetic correctly is key.In our given example, we used the formula twice:

• Midpoint of segment \( AB \): \( \left( \frac{3}{2}, \frac{9}{2} \right) \)
• Midpoint of segment \( CD \): \( \left( \frac{7}{2}, \frac{11}{2} \right) \)

These midpoints become the new points of interest for any further analysis, like measuring the distance between them.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry to solve problems involving points, lines, and shapes. It enables precise calculations and visual representation of geometric concepts using the coordinate plane.This field uses a Cartesian plane where every point is identified by an ordered pair \((x, y)\). We leverage algebraic techniques to explore geometric properties and relationships. This method allows us to calculate distances, angles, and other useful geometric properties.A strong grasp of coordinate geometry principles enables us to:

• Describe positions using coordinates.
• Determine distances using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Find midpoints to understand geometric relationships.

It's a versatile and powerful tool for solving complex geometric problems with the simplicity of linear equations and coordinates.The exercise highlights how coordinate geometry allows us to calculate and find essential geometric information like the distance between midpoints using these coordinate principles.

Line Segment

A line segment is one of the fundamental concepts in geometry, representing the portion of a line that exists between two endpoints. Unlike lines that extend infinitely in either direction, line segments have a definite start and end.Consider a line segment \( AB \), defined by the endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \). This segment represents all the points in the line between \( A \) and \( B \). These are the crucial aspects of understanding line segments:

• Each segment can be described using its endpoints.
• The length or distance can be calculated using the distance formula.
• The midpoint provides the center of the segment.

Understanding line segments help in solving various geometric problems, allowing us to dissect shapes, find centroids, or even solve real-life problems involving distances.The exercise involved finding the line segment joining the midpoints of two other segments. This concept applies the knowledge of line segments in a more complex situation by focusing on relationship building rather than simple length determination.