Problem 98 - A Graphical Approach to Precalculus with Limits

Question

Suppose that \(P\) is an endpoint of a segment \(P Q\) 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

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

Step 1: Identify the midpoint formula

The midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). In this problem, the midpoint \(M\) is given as \((-2, -4)\).

Step 2: Set up equations for the coordinates of Q

We know the coordinates of \(P = (13, 5)\) and \(M = (-2, -4)\). Let \(Q = (x, y)\). According to the midpoint formula, we have:\[ \frac{13 + x}{2} = -2 \] and \[ \frac{5 + y}{2} = -4 \].

Step 3: Solve for x-coordinate of Q

Solve the first equation \(\frac{13 + x}{2} = -2\). Multiply both sides by 2 to get \(13 + x = -4\). Subtract 13 from both sides to find \(x = -17\).

Step 4: Solve for y-coordinate of Q

Solve the second equation \(\frac{5 + y}{2} = -4\). Multiply both sides by 2 to obtain \(5 + y = -8\). Subtract 5 from both sides to solve for \(y = -13\).

Step 5: Determine the coordinates of Q

After solving the equations, the coordinates of \(Q\) are found to be \((-17, -13)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry to provide a deeper understanding of points, lines, and shapes in a coordinate plane. The coordinate plane itself consists of two axes: the horizontal x-axis and the vertical y-axis. Where these two axes intersect is known as the origin, labeled as the point \(0, 0\). This system allows us to establish precise locations for points using ordered pairs of numbers, such as \(13, 5\) or \(-2, -4\).

One of the essential concepts in coordinate geometry is the distance between points and how to find the midpoint of a line segment connecting two points. The midpoint provides a unique point that is equidistant from the endpoints of the segment. This is especially useful in many geometric constructions and problems, where understanding and manipulating the position of lines and points can yield visual proofs and solutions.

The formula for calculating the midpoint is \((x_1 + x_2)/2, (y_1 + y_2)/2\). This formula effectively averages the x-coordinates and y-coordinates of two points. In the exercise provided, the midpoint \((-2, -4)\) helps us find another endpoint \(Q\) when combined with one known endpoint \(P\). Understanding the basics of coordinate geometry is key to solving such problems efficiently.

Finding Coordinates

Finding coordinates in a coordinate plane involves determining the exact numerical representation of a position. Often, you'll have some data, like one endpoint and a midpoint, and need to find the coordinates of the remaining endpoint.

In the current problem, we have the first endpoint \(P(13,5)\) and the midpoint \(M(-2,-4)\). To find the coordinates of the second endpoint \(Q(x, y)\), we use the midpoint formula that integrates both the known point and the midpoint to create equations for the unknown coordinates. These equations are calculated as follows:

• For the x-coordinate: '\((x_1 + x)/2 = x_m\)' translates into '\((13 + x)/2 = -2\)'
• For the y-coordinate: '\((y_1 + y)/2 = y_m\)' translates into '\((5 + y)/2 = -4\)'

This setup ensures that the computed midpoint indeed corresponds to the provided coordinates \((-2, -4)\). The careful handling of average values is crucial while organizing such expressions to successfully find unknown locations in this mathematical realm.

Equation Solving

Equation solving is a fundamental mathematical process critical in identifying unknown values from given relationships or conditions. In our problem, we need to find the values of \(x\) and \(y\) for the endpoint \(Q\) using the equations derived from the midpoint formula. This involves solving for each coordinate independently using basic algebra.

• **For the x-coordinate**, solve \(\frac{13 + x}{2} = -2\):
  • Multiply both sides of the equation by 2 to eliminate the fraction, resulting in \13 + x = -4\.
  • Subtract 13 from both sides, giving \x = -17\.
• **For the y-coordinate**, solve \(\frac{5 + y}{2} = -4\):
  • Multiply both sides by 2, leading to \5 + y = -8\.
  • Subtract 5, resulting in \y = -13\.

After solving these simple linear equations, the coordinates of endpoint \(Q\) are determined to be \((-17, -13)\). Understanding how to manipulate and solve these equations allows you to extract exact values needed in various geometric contexts.