Problem 56 - College Algebra

Question

Given these four points: \(A(1,3), B(-3,5), C(4,7)\) and \(D(5,-4),\) find the coordinates of the midpoint of line segments \(\overline{A B}\) and \(\overline{C D}\) .

Step-by-Step Solution

Answer

Midpoint of \(\overline{AB}\): (-1, 4); Midpoint of \(\overline{CD}\): (4.5, 1.5).

Step 1: Understand the Midpoint Formula

The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the midpoint formula: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \] We will use this formula to find the midpoints of segments \(\overline{AB}\) and \(\overline{CD}\).

Step 2: Apply the Midpoint Formula to Segment AB

To find the midpoint of segment \(\overline{AB}\), use the coordinates of points \(A(1,3)\) and \(B(-3,5)\):- Substitute into the formula as follows: \[ \left( \frac{1 + (-3)}{2}, \frac{3 + 5}{2} \right) = \left( \frac{-2}{2}, \frac{8}{2} \right) = (-1, 4). \] So, the midpoint of \(\overline{AB}\) is \((-1, 4)\).

Step 3: Apply the Midpoint Formula to Segment CD

To calculate the midpoint of segment \overline{CD}, use the coordinates of points \(C(4,7)\) and \(D(5,-4)\):- Substitute into the midpoint formula: \[ \left( \frac{4 + 5}{2}, \frac{7 + (-4)}{2} \right) = \left( \frac{9}{2}, \frac{3}{2} \right) = \left( 4.5, 1.5 \right). \] Therefore, the midpoint of \overline{CD}\ is \(4.5, 1.5\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic formulas to solve geometrical problems. It provides a way to describe and analyze the positions and shapes of geometrical figures using a coordinate system. This can be particularly useful when dealing with shapes, lines, and points on a plane.

The main idea behind coordinate geometry is to use ordered pairs of numbers, called coordinates, to denote locations on a plane. Each point in the plane is represented by a pair \(x, y\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate.

• Coordinate geometry helps to find distances between points, angles between lines, and also equations of lines and curves.
• It is extensively used in various fields like physics, engineering, and computer graphics.

By understanding the principles of coordinate geometry, solving problems that involve line segments, midpoints, and distances between points becomes significantly easier.

Line Segments

In coordinate geometry, a line segment is a part of a line that is bounded by two distinct end points. Unlike a line, which extends infinitely in both directions, a line segment has a definite beginning and end. Understanding line segments is crucial for calculating midpoints and distances within a plane.

Line segments are typically represented by the endpoints with a bar over the labels, such as \(\overline{AB}\) for the segment with endpoints A and B.

To find points or features like midpoints on a line segment, the coordinates of its endpoints are used:

• The length of a line segment can be calculated using the distance formula: \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\.
• A line segment's location and properties help detail its role in shapes and figures.

Overall, understanding line segments in coordinate geometry is fundamental to interpreting and solving geometrical problems effectively.

Midpoint Calculation

The midpoint of a line segment is the point that divides the segment into two equal parts. The Midpoint Formula is a simple yet powerful tool in coordinate geometry. It enables the calculation of this central point using the coordinates of the segment's endpoints.

The formula for finding the midpoint \(M\) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \]

This formula essentially averages the x-coordinates and y-coordinates separately to find the middle. Let’s outline the steps needed to apply it:

• Identify the coordinates of the endpoints.
• Add the x-coordinates and divide by 2 to get the x-coordinate of the midpoint.
• Add the y-coordinates and divide by 2 to get the y-coordinate of the midpoint.

This straightforward method provides an accurate midpoint, which is crucial when further analyzing geometric figures or solving coordinate geometry problems.