Problem 95 - 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 \(Q\). $$P(13 x,-23 x), Q(6 x, x), x>0$$

Step-by-Step Solution

Answer

Distance: \(25x\); Midpoint: \(\left(\frac{19x}{2}, -11x\right)\)."

Step 1: Calculate the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. For the points \(P(13x, -23x)\) and \(Q(6x, x)\), substitute \((x_1, y_1) = (13x, -23x)\) and \((x_2, y_2) = (6x, x)\).

Step 2: Substitute Coordinates into Distance Formula

Substitute the coordinates of \(P\) and \(Q\) into the distance formula: \[ d = \sqrt{(6x - 13x)^2 + (x + 23x)^2} \]. Simplify the expressions: \[ d = \sqrt{(-7x)^2 + (24x)^2} \].

Step 3: Simplify the Distance Expression

Further simplify: \[ d = \sqrt{49x^2 + 576x^2} = \sqrt{625x^2} \]. Since \(x > 0\), \(d = 25x\).

Step 4: Distance Between Points

Thus, the distance between the points \(P\) and \(Q\) is \(25x\).

Step 5: Find the Midpoint Formula

The midpoint \((x_m, y_m)\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[(x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\].

Step 6: Substitute Coordinates into Midpoint Formula

Substitute the coordinates of \(P(13x, -23x)\) and \(Q(6x, x)\) into the midpoint formula: \[(x_m, y_m) = \left(\frac{13x + 6x}{2}, \frac{-23x + x}{2}\right)\]. Simplify the expressions: \[(x_m, y_m) = \left(\frac{19x}{2}, \frac{-22x}{2}\right)\].

Step 7: Simplify the Midpoint Expression

Simplify further to get the coordinates of the midpoint: \((x_m, y_m) = \left(\frac{19x}{2}, -11x\right)\).

Step 8: Coordinates of Midpoint

Thus, the coordinates of the midpoint \(M\) are \(\left(\frac{19x}{2}, -11x\right)\).

Key Concepts

Distance Formula

The distance formula is a magical tool in geometry that lets you find the space between two points on a coordinate plane. Imagine you have two points, \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), and you want to know how far they are from each other. The formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].

Think of the formula as a way to create a "straight line" bridge over the graph's grid. It uses the Pythagorean Theorem principles, where the distance forms the hypotenuse of a right triangle. Here's the breakdown:

• The term \((x_2 - x_1)^2\) measures the horizontal distance, squared.
• The term \((y_2 - y_1)^2\) measures the vertical distance, squared.

The square root then completes the Pythagorean relationship to give you the direct distance. In the exercise, substituting for points \(P(13x, -23x)\) and \(Q(6x, x)\) yielded \(d = 25x\), which is the calculated distance after simplification.

Midpoint Formula

The midpoint formula helps find the exact center between two points on a line segment. It’s like finding the halfway mark on a piece of string. The formula to find the midpoint \((x_m, y_m)\) is given by: \[ (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \].

With this formula, it sums up the coordinates of two points and divides by 2, effectively finding the average. Here’s how it works step-by-step:

• Add the x-coordinates of the two points and divide by 2 to get \((x_1 + x_2)/2\).
• Add the y-coordinates and do the same to get \((y_1 + y_2)/2\).

Following these steps with the exercise's points \(P(13x, -23x)\) and \(Q(6x, x)\), you achieve a midpoint at \(\left(\frac{19x}{2}, -11x\right)\). This means \(M\) is positioned perfectly in the middle between \(P\) and \(Q\).

Coordinates

Coordinates are the backbone of geometry on a plane. They describe a point's exact location using two values. In a two-dimensional plane, coordinates come in pairs known as \(x\) and \(y\). For example, when we say point \(P(3, 4)\), it means:

• The x-coordinate is 3 and shows how far along the horizontal axis (left-right) the point is.
• The y-coordinate is 4 and indicates how far up or down (vertical axis) it is.

Coordinates on a plane are typically visualized on a grid, with each point having a specific pair of \(x\) and \(y\) positions. In our exercise, the points \(P(13x, -23x)\) and \(Q(6x, x)\) show how coordinates multiply by a factor \(x\). This means their exact position can vary based on \(x\) but the relative geometry remains the same. Understanding coordinates is key to using both the distance and midpoint formulas effectively as it depicts the critical "starting spots" for any calculations.