Problem 66 - Elementary and Intermediate Algebra

Question

Solve each problem. If \(\left(\frac{1}{2},-2\right)\) is the midpoint of segment \(Q P\) and the coordinates of \(P\) are \(\left(-\frac{5}{2}, 5\right),\) find the coordinates of \(Q .\)

Step-by-Step Solution

Answer

The coordinates of \( Q \) are \( \left( \frac{7}{2}, -9 \right) \).

Step 1: Understanding the Midpoint Formula

The formula for the midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]In this problem, the midpoint is \( \left( \frac{1}{2}, -2 \right) \), and the coordinates of \( P \) are \( \left( -\frac{5}{2}, 5 \right) \). Our goal is to find the coordinates of \( Q \), which we will denote as \( (x, y) \).

Step 2: Apply the Midpoint Formula for the x-coordinate

Using the midpoint formula for the x-coordinate, we have:\[ \frac{-\frac{5}{2} + x}{2} = \frac{1}{2} \]Next, solve for \( x \) by clearing the fraction and isolating \( x \).

Step 3: Solving for x-coordinate

Multiply both sides of the equation by 2 to clear the denominator:\[ -\frac{5}{2} + x = 1 \]To isolate \( x \), add \( \frac{5}{2} \) to both sides:\[ x = 1 + \frac{5}{2} \]Convert 1 to a fraction with a denominator of 2 for easy addition:\[ x = \frac{2}{2} + \frac{5}{2} = \frac{7}{2} \]

Step 4: Apply the Midpoint Formula for y-coordinate

Using the midpoint formula for the y-coordinate, we have:\[ \frac{5 + y}{2} = -2 \]Solve for \( y \) by clearing the fraction and isolating \( y \).

Step 5: Solving for y-coordinate

Multiply both sides of the equation by 2 to clear the denominator:\[ 5 + y = -4 \]To isolate \( y \), subtract 5 from both sides:\[ y = -4 - 5 = -9 \]

Step 6: Write the Coordinates of Q

The coordinates of \( Q \) are \( \left( \frac{7}{2}, -9 \right) \). We found these by solving for \( x \) and \( y \) using the midpoint formula.

Key Concepts

Coordinate Geometry

In coordinate geometry, points are placed in a 2D space that is defined by a horizontal (x) axis and a vertical (y) axis. This system allows us to identify each location with a unique pair of numbers known as coordinates, written as \( (x, y) \). Understanding coordinate geometry is fundamental for solving problems involving distances and midpoints on a plane.

The midpoint formula, which is used to find the center point between two coordinates, is a crucial concept. If you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint is calculated as:

• \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

In the given exercise, one endpoint (\(P\)) and the midpoint are known, enabling you to determine the missing endpoint using this formula.

Coordinate geometry not only involves finding midpoints but also exploring various geometric properties and relationships within this flat, two-dimensional plane.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when plotted on a coordinate plane. They are typically in the form of \( y = mx + c \), where \( m \) stands for slope and \( c \) is the y-intercept.

In problem scenarios involving endpoints and midpoints, linear equations help in formulating relationships between different components of the line segment. For instance, using the midpoint formula involves assuming linear relationships between the coordinates. Through algebraic manipulation of linear equations, such as solving for \( x \) and \( y \), these problems can be strategically tackled.

This exercise requires solving two linear expressions, one for each coordinate, to find the unknown endpoint of a line segment. The balance of an equation effectively illustrates how modifying one variable affects another, embodying the concept of equality essential in algebra.

Problem Solving in Algebra

Problem-solving in algebra involves using mathematical operations and formulas to find unknown values or solve equations. In this problem, you use logical reasoning and systematic methods—central components of algebra—to identify the unknown coordinates for the point \( Q \).

The process begins with understanding the given data and identifying which algebraic operations will lead to a solution. For instance, you need to isolate \( x \) or \( y \) on one side of the equation using the midpoint formula. Here are the steps taken in the exercise:

• Set up the correct equation using a known midpoint and one endpoint.
• Solve for one variable by isolating it, using techniques such as clearing denominators and combining like terms.
• Repeat this process for the second variable.

This systematic approach breaks down complex problems into manageable tasks, proving the power of algebraic problem-solving in mathematical challenges. By the end of the solution, you identify the coordinates of \( Q \), showcasing how algebra assists in uncovering unknown components of a problem.