Problem 98 - A Graphical Approach to College Algebra

Question

Suppose that P is an endpoint of a segment PQ and M is the midpoint of $P Q . Find the coordinates of endpoint Q. $$P(13,5), M(-2,-4)$$

Step-by-Step Solution

Answer

Endpoint Q is at (-17, -13).

Step 1: Understand the Midpoint Formula

The midpoint \( M \) of a segment with endpoints \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is given by:\[ M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Here, \( P(13, 5) \) and \( M(-2, -4) \) are the given points.

Step 2: Set Up Equations for the X-Coordinate

Use the x-coordinate of the midpoint formula:\[ -2 = \frac{13 + x_2}{2} \]Solve for \( x_2 \).

Step 3: Solve for X-Coordinate of Q

Multiply both sides by 2 to eliminate the fraction:\[ -4 = 13 + x_2 \]Subtract 13:\[ x_2 = -17 \]

Step 4: Set Up Equations for the Y-Coordinate

Next, use the y-coordinate of the midpoint formula:\[ -4 = \frac{5 + y_2}{2} \]Solve for \( y_2 \).

Step 5: Solve for Y-Coordinate of Q

Multiply both sides by 2:\[ -8 = 5 + y_2 \]Subtract 5:\[ y_2 = -13 \]

Step 6: Conclude the Coordinates of Q

The coordinates of endpoint \( Q \) are \( (-17, -13) \).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that enables us to describe and analyze the position of points using numbers. It blends algebra and geometry to solve geometric problems, providing a way to graphically represent and solve equations visually. This is particularly useful to determine properties like distance and midpoint.
Within coordinate geometry, every point on a plane is defined by an ordered pair \(x, y\) where \x\ is the horizontal coordinate and \y\ is the vertical coordinate.
The essential aspect of coordinate geometry is that it allows us to examine geometric figures using a coordinate system, which is handy for calculating distances, midpoints, area, and transformations.

Endpoints of a Segment

In coordinate geometry, a segment has two endpoints, which are its defining points. Think of a segment as a straight path connecting two distinct points in a geometric space. Each endpoint is represented by a pair of coordinates.
These endpoints serve as the boundary markers of a segment. They determine the extent and position of the segment on the coordinate plane. For instance, in our exercise, point \(P(13, 5)\) is one of the endpoints, while another endpoint \(Q(x_2, y_2)\) needs to be found given the midpoint.
Understanding these endpoints and their roles helps in constructing geometric shapes and assists in calculations like finding the midpoint or distance.

Midpoint Calculations

The midpoint of a segment is the point that divides the segment into two equal parts. The midpoint formula provides a means to calculate this point if the endpoints are known, or vice versa, to find an unknown endpoint if the midpoint and one endpoint are given.
The formula for the midpoint \(M\) of two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is:

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

In our exercise, the midpoint \(M(-2, -4)\) and endpoint \(P(13, 5)\) are provided. Using the formula, equations are set up to solve for the unknown coordinates of \(Q\). This involves algebraic manipulation to find these coordinates representative of \(Q\).

Algebraic Problem Solving

Algebraic problem solving involves using algebra to manipulate and solve equations derived from geometric formulas. This method allows for finding unknown values through established algebraic procedures.
In solving for the coordinates of \(Q\), once the midpoint formula is applied, algebra is used to solve the resulting equations.
Here’s a breakdown of solving for \(x_2\):

• Set up the equation: \(-2 = \frac{13 + x_2}{2}\)
• Multiply through by 2 to clear the fraction: \(-4 = 13 + x_2\)
• Isolate \(x_2\) by subtracting 13: \(x_2 = -17\)

Similarly, solve for \(y_2\) to find its coordinate making use of the same steps, ultimately resulting in the coordinates \((-17, -13)\) for endpoint \(Q\). Algebraic techniques like these enable clear, logical paths to solutions that would otherwise be challenging to find.