Problem 56 - College Algebra with Corequisite Support

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 \(\overline{\mathrm{AB}}\) and \(\overline{\mathrm{CD}} .\)

Step-by-Step Solution

Answer

Midpoint of AB is \\((-1, 4)\\) and midpoint of CD is \\(4.5, 1.5\\)\.

Step 1: Understanding Midpoints

The midpoint of a line segment is the point that is exactly halfway between the two endpoints. To find the midpoint, we average the x-coordinates of the endpoints and the y-coordinates of the endpoints. The formula for the midpoint, given points \((x_1, y_1)\) and \((x_2, y_2)\), is: \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)\.

Step 2: Finding Midpoint of AB

To find the midpoint of \(\overline{\mathrm{AB}}\), use the coordinates of points A \(1,3\) and B \(-3,5\). Substitute into the midpoint formula: \((\frac{1 + (-3)}{2}, \frac{3 + 5}{2}) = (\frac{-2}{2}, \frac{8}{2}) = (-1, 4)\)\.

Step 3: Finding Midpoint of CD

Now, calculate the midpoint of \(\overline{\mathrm{CD}}\), using points C \(4,7\) and D \(5,-4\). Insert these into the formula: \((\frac{4 + 5}{2}, \frac{7 + (-4)}{2}) = (\frac{9}{2}, \frac{3}{2}) = (4.5, 1.5)\)\.

Key Concepts

Coordinates

When we talk about coordinates in geometry, we're referring to a set of values that show an exact position on a plane. These values are usually in the form of \((x, y)\), where \(x\) represents the horizontal position and \(y\) represents the vertical position. Whenever we describe points, line segments, or even larger shapes, coordinates are essential. They help us pinpoint specific locations.

The method we use most often to describe these points is the Cartesian coordinate system.

• The x-coordinate tells us how far left or right a point is from the origin, which is \((0,0)\).
• The y-coordinate tells us how far up or down.

By using coordinates like \(A(1, 3)\), we're easily guiding ourselves to a precise spot on our grid.

Line Segments

A line segment is a part of a line that has two distinct endpoints. Unlike a line that stretches infinitely, a line segment connects these two points and has a measurable length.

When we speak about the line segment \(\overline{AB}\), we're looking at a straight path from point A to point B, including both endpoints. Here’s how line segments are defined:

• Endpoints: These are the two points that mark the boundary of the segment, like \((1, 3)\) and \((-3, 5)\) for \(\overline{AB}\).
• Length: The distance between the two endpoints can be calculated, although it’s not required for finding the midpoint.

Line segments are fundamental in geometry, connecting points and forming shapes when multiple segments come together.

Averaging Points

The concept of averaging is used to find the midpoint of a line segment. By averaging, we identify a point that is equidistant from both endpoints.The formula for finding the midpoint is: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]When applied, averaging gives us the coordinates of the midpoint. This essentially tells us:

• By combining the x-coordinates and halving, we find a middle value for \(x\).
• Similarly, by doing the same for the y-coordinates, we average for \(y\).

This technique is pivotal in geometry, providing a balanced location between two defined points and commonly used for dividing segments equally or analyzing symmetrical properties.