Problem 99 - 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 \(Q\). $$P(5.64,8.21), M(-4.04,1.60)$$

Step-by-Step Solution

Answer

The coordinates of endpoint \( Q \) are \((-13.72, -5.01)\).

Step 1: Understanding Midpoint Formula

The midpoint formula is used to find the position of the midpoint \( M \) of a line segment when the coordinates of the endpoints \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) are provided. The formula is: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Step 2: Setup Equations for Coordinates

Since we already have \( P(5.64,8.21) \) and \( M(-4.04,1.60) \) and need to find \( Q(x_2, y_2) \), we can use the midpoint formula to set two equations for the \( x \) and \( y \) coordinates: \( -4.04 = \frac{5.64 + x_2}{2} \) and \( 1.60 = \frac{8.21 + y_2}{2} \).

Step 3: Solve for \(x_2\)

We solve \(-4.04 = \frac{5.64 + x_2}{2}\). Multiply both sides by 2 to eliminate the fraction: \( -8.08 = 5.64 + x_2 \). Subtract 5.64 from both sides: \( x_2 = -8.08 - 5.64 = -13.72 \).

Step 4: Solve for \(y_2\)

Next, solve \(1.60 = \frac{8.21 + y_2}{2}\). Multiply both sides by 2: \(3.20 = 8.21 + y_2\). Subtract 8.21 from both sides: \( y_2 = 3.20 - 8.21 = -5.01 \).

Step 5: Solution Verification

Verify by plugging back the values of \(x_2\) and \(y_2\) into the midpoint formula. Check \( M = \left( \frac{5.64 + (-13.72)}{2}, \frac{8.21 + (-5.01)}{2} \right) = (-4.04, 1.60) \), confirming calculations are correct.

Key Concepts

Coordinate Geometry

Coordinate Geometry is the study of geometric figures using a coordinate system. In this particular exercise, we work with points, lines, and their interactions in a plane. Each point is described using coordinates \((x, y)\), which tell you its position on the plane. To find the missing point in a line segment when given one endpoint and the midpoint, we employ the Midpoint Formula. This formula helps us calculate the average coordinates of two points, showing how the midpoint divides the line segment equally. It's crucial for understanding how distances and midpoints work in real-world applications and other math problems.

Equation Solving

Equation Solving is a fundamental part of mathematics that allows us to find unknown variables. In this exercise, we use equation solving to find the coordinates of the missing endpoint \(Q\). We begin by setting up equations based on the midpoint formula: \(-4.04 = \frac{5.64 + x_2}{2}\) and \(1.60 = \frac{8.21 + y_2}{2}\).

By rearranging these equations, we can solve for each variable one step at a time:

• Multiply both sides by 2 to get rid of the fractions.
• Solve the resulting simple equations for \(x_2\) and \(y_2\).

The solution gives \(x_2 = -13.72\) and \(y_2 = -5.01\). These values, when substituted back into the midpoint formula, confirm the accuracy of our solution.

Precalculus Solutions

Precalculus is a course that bridges the gap between algebra and calculus by covering concepts such as functions, coordinates, and equations. In our task, Understanding the midpoint in the context of coordinate geometry is a stepping stone in precalculus.

This exercise demonstrates how to apply algebraic manipulation to geometric contexts, preparing students for more advanced calculus concepts.

• Learning to find coordinates involves recognizing patterns and relationships between numbers.
• Practicing these problems enhances problem-solving skills, particularly how to handle equations with multiple steps.

The process of calculating unknown coordinates through established formulas increases your readiness for tackling further precalculus challenges. So, mastering these steps now will significantly assist in understanding more complex mathematical concepts later on.