Problem 91 - A Graphical Approach to Precalculus with Limits

Question

Find (a) the distance between \(P\) and \(Q\) and (b) the coordinates of the midpoint \(M\) of the segment joining \(P\) and \(Q\). $$P(-8,-2), Q(-3,-5)$$

Step-by-Step Solution

Answer

(a) Distance: approximately 5.83 units. (b) Midpoint: \((-5.5, -3.5)\).

Step 1: Calculate the Distance Using the Distance Formula

The distance between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates for \( P(-8, -2) \) and \( Q(-3, -5) \) into the formula, we get: \[ d = \sqrt{(-3 + 8)^2 + (-5 + 2)^2} \] \[ d = \sqrt{5^2 + (-3)^2} \] \[ d = \sqrt{25 + 9} \] \[ d = \sqrt{34} \] \[ d \approx 5.83 \]

Step 2: Calculate the Midpoint Using the Midpoint Formula

The midpoint \( M \) of a line segment joining points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is calculated using the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] For points \( P(-8, -2) \) and \( Q(-3, -5) \), substitute their coordinates: \[ M = \left( \frac{-8 + (-3)}{2}, \frac{-2 + (-5)}{2} \right) \] \[ M = \left( \frac{-11}{2}, \frac{-7}{2} \right) \] \[ M = \left( -5.5, -3.5 \right) \]

Key Concepts

Midpoint Formula

The midpoint formula is a powerful tool in coordinate geometry to find the exact middle point between two points on a plane. Imagine you have two endpoints of a line segment, represented as coordinates

• \( P(x_1, y_1) \)
• \( Q(x_2, y_2) \)
.

The midpoint, \( M(x, y) \), lies exactly halfway along the line connecting \( P \) and \( Q \) both horizontally and vertically.
It is given by the formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Here's how it works in practice: let's take our points \( P(-8, -2) \) and \( Q(-3, -5) \). To find the x-coordinate of the midpoint \( M \), add the x-coordinates of \( P \) and \( Q \), which is \(-8 + (-3) = -11\). Divide this sum by 2 to get the x-value, resulting in \(-5.5\).
Similarly, for the y-coordinate, add \(-2 + (-5) = -7\), then divide by 2 to get \(-3.5\).
Thus, the midpoint \( M \) is \((-5.5, -3.5)\).
This formula ensures we equally split the line segment into two equal parts.

Distance Between Points

The distance between two points in coordinate geometry is a measure of the straight line connecting them. We use the distance formula, which comes from the Pythagorean Theorem, to calculate this.
Given two points:

• \( P(x_1, y_1) \)
• \( Q(x_2, y_2) \)

The distance \( d \) between them is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Let's look at our specific example where \( P = (-8, -2) \) and \( Q = (-3, -5) \).
First, find the difference in x-coordinates: \(-3 - (-8) = 5\).
Then, find the difference in y-coordinates: \(-5 - (-2) = -3\).
Next, square these differences: \(5^2 = 25\) and \((-3)^2 = 9\).
Adding these squared differences gives \(25 + 9 = 34\).
Taking the square root provides the distance: \(d = \sqrt{34} \approx 5.83\).
This distance tells us the length directly between \( P \) and \( Q \) without needing to consider directions or other paths.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to describe and analyze geometric figures. This branch of mathematics allows us to use equations to represent geometric shapes on the Cartesian plane.Coordinate geometry uses ordered pairs of numbers to determine positions and solve geometrical problems. Let's understand some fundamental elements:

• Points: Each point is represented by an ordered pair \((x, y)\).
• Lines: Can be described using linear equations and their slopes.
• Figures: Shapes like triangles and circles have equations that outline their properties.

With coordinate geometry, you can solve problems like finding distances (as we did with the distance formula) and midpoints (using the midpoint formula). These tools let you understand distances and relative positions of points and shapes in a more precise and mathematical way.
In essence, coordinate geometry opens a pathway to rigorously explore the spatial relationships between points and structures on a plane, blending the intuition of geometry with the precision of algebra.