Problem 56 - College Algebra

Question

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

Step-by-Step Solution

Answer

The midpoints are \((-1, 4)\) for \(\overline{AB}\) and \((4.5, 1.5)\) for \(\overline{CD}\).

Step 1: Identify Midpoint Formula

To find the midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), use the formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).

Step 2: Apply Formula to \(\overline{AB}\)

For \(\overline{AB}\), the endpoints are \(A(1, 3)\) and \(B(-3, 5)\). Using the midpoint formula:\[\left( \frac{1 + (-3)}{2}, \frac{3 + 5}{2} \right) = \left( \frac{-2}{2}, \frac{8}{2} \right) = (-1, 4)\].

Step 3: Apply Formula to \(\overline{CD}\)

For \(\overline{CD}\), the endpoints are \(C(4, 7)\) and \(D(5, -4)\). Using the midpoint formula:\[\left( \frac{4 + 5}{2}, \frac{7 + (-4)}{2} \right) = \left( \frac{9}{2}, \frac{3}{2} \right) = \left(4.5, 1.5\right)\].

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to study geometric shapes and measurements using the coordinate plane. The plane is a two-dimensional space where each point is defined by a pair of numbers, typically in the form \(x, y\). The coordinate plane consists of two perpendicular lines, known as the x-axis (horizontal) and y-axis (vertical), which intersect at the origin \(0, 0\).

Using this system, we can easily determine the location of any point in the plane by its coordinates. Understanding coordinate geometry is fundamental for analyzing figures, calculating distances, and finding midpoints, which is crucial in solving geometric problems.

In this context, the midpoint formula, which averages the x and y coordinates of two endpoints of a line segment, is an essential tool. This formula helps us identify the exact center point of a line segment in the coordinate plane.

Line Segments

In coordinate geometry, a line segment is a part of a line that is bounded by two distinct endpoints. It contains all the points on the line between these two endpoints. Each endpoint can be represented with coordinates, such as \(A(1,3)\) and \(B(-3,5)\) for line segment \(\overline{AB}\).

To analyze line segments in the coordinate plane, it is important to understand their properties:

• Length: The physical length of the segment is determined by the distance formula.
• Midpoint: The center of the line segment, found by averaging the coordinates of the endpoints.

The midpoint tells us the exact location that is equidistant from both endpoints, effectively dividing the segment into two equal halves. This concept is frequently utilized in various mathematical and real-world applications, such as finding the point of balance or symmetry.

Algebraic Expressions

Algebraic expressions in coordinate geometry often involve variables that represent the coordinates of points. For example, when using the midpoint formula, \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\), the expression becomes a tool for calculating the midpoint of a line segment.

These expressions allow us to manipulate and solve for unknown quantities effectively:

• Simplification: Combining like terms to simplify expressions.
• Substitution: Plugging in numerical values to evaluate expressions.

In the solution, using algebraic expressions allows for the substitution of the endpoints' coordinates to find the midpoints \((-1, 4)\) and \(4.5, 1.5\). Mastering these expressions is crucial for translating geometric problems into algebraic form, which simplifies the process of finding solutions efficiently.