Problem 13 - Precalculus Mathematics for Calculus

Question

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them. $$(0,8),(6,16)$$

Step-by-Step Solution

Answer

(0,8) and (6,16) have a distance of 10 units and a midpoint of (3,12).

Step 1: Plot the Points

First, identify the coordinates of the given points. The points are \((0,8)\) and \((6,16)\). On a coordinate plane, locate and mark (0, 8) on the y-axis, which is horizontal to 8 units up from the origin. Then locate and mark (6, 16) by moving 6 units to the right along the x-axis and 16 units up from the origin.

Step 2: Calculate the Distance

To find the distance between the points \((x_1, y_1) = (0, 8)\) and \((x_2, y_2) = (6, 16)\), use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].Plug in the values: \[ d = \sqrt{(6 - 0)^2 + (16 - 8)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \].Thus, the distance between the points is 10 units.

Step 3: Calculate the Midpoint

To find the midpoint of the segment joining the points \((0, 8)\) and \((6, 16)\), use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].Substitute the coordinates into the formula: \[ M = \left( \frac{0 + 6}{2}, \frac{8 + 16}{2} \right) = \left( \frac{6}{2}, \frac{24}{2} \right) = (3, 12) \].Thus, the midpoint of the segment is \((3, 12)\).

Key Concepts

Midpoint Formula

The midpoint formula allows us to find the middle point between two coordinates on a plane. This is super useful for dividing a line segment into two equal parts. Imagine you've got two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), and you want to find what's right in the middle. This midpoint is found using the formula:

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

Here's how it works with our example: our points are at \( (0, 8) \) and \( (6, 16) \). Plug these into the formula, and we find:

• \[ M = \left( \frac{0 + 6}{2}, \frac{8 + 16}{2} \right) = (3, 12) \]

So, the midpoint of the line segment connecting these two points is \( (3, 12) \). This coordinate is exactly halfway between the two original points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebraic methods with geometrical problems. By using a coordinate plane, it allows us to understand the spatial relationships of points. This is fundamental to finding distances and midpoints between points.A coordinate plane consists of two axes:

• The x-axis (horizontal line)
• The y-axis (vertical line)

To locate a point, you need a pair of numbers, like \( (x, y) \). The first number \( x \) represents how far to move left or right from the origin, and the second number \( y \) shows how far up or down to move.
In our example, to plot \( (0, 8) \), you don't move left or right, just 8 units up. For \( (6, 16) \), move 6 units to the right and 16 units up.This structured way of positioning points helps us easily calculate other features, like distances or midpoints.

Plotting Points

Plotting points is all about identifying a specific location on the coordinate plane using coordinates. For each point, we use an ordered pair \( (x, y) \) to determine its position. Let’s break it down a bit further:Follow these simple steps to plot points:

• Start from the origin \( (0, 0) \), which is where the x-axis and y-axis meet.
• Move horizontally by \( x \) units. Right for positive values, left for negative.
• Then move vertically by \( y \) units. Up for positive values, down for negative.

To plot the point \( (0, 8) \), start at the origin, stay on the y-axis (no x movement), and go up 8 units. For \( (6, 16) \), go 6 units right, then 16 units up from the origin.Visually plotting these points helps form a line segment, making it easier to work with midpoint and distance calculations. Imagine them as landmarks, allowing you to "see" the problem more clearly.