Problem 17 - 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. $$(6,-2),(-6,2)$$

Step-by-Step Solution

Answer

(a) Plot at (6,-2) and (-6,2); (b) Distance = \(4\sqrt{10}\); (c) Midpoint = (0,0).

Step 1: Plot the Points

Plot the points \((6, -2)\) and \((-6, 2)\) on a coordinate plane. The first point, \((6, -2)\), is located 6 units to the right of the origin and 2 units down. The second point, \((-6, 2)\), is located 6 units to the left of the origin and 2 units up.

Step 2: Apply the Distance Formula

The distance formula is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute \((x_1, y_1) = (6, -2)\) and \((x_2, y_2) = (-6, 2)\):\[d = \sqrt{(-6 - 6)^2 + (2 + 2)^2} = \sqrt{(-12)^2 + 4^2} = \sqrt{144 + 16} = \sqrt{160} = 4\sqrt{10}\]Thus, the distance between the points is \(4\sqrt{10}\).

Step 3: Apply the Midpoint Formula

The midpoint formula is given by \((M_x, M_y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute \((x_1, y_1) = (6, -2)\) and \((x_2, y_2) = (-6, 2)\):\[M_x = \frac{6 + (-6)}{2} = 0, \quad M_y = \frac{-2 + 2}{2} = 0\]The midpoint of the segment that joins the points is \((0, 0)\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where points are defined by a pair of numerical coordinates. These coordinates are written as

• **x-coordinate**: The position of the point along the horizontal axis (usually called the x-axis).
• **y-coordinate**: The position of the point along the vertical axis (usually called the y-axis).

The origin, where both x and y are zero, is the starting point of both axes and is located at (0, 0). Every point on the plane can be represented as a pair of numbers. For example, the point (6, -2) is located:

• 6 units right of the origin on the x-axis.
• 2 units down from the origin on the y-axis.

Meanwhile, the point (-6, 2) is located 6 units left on the x-axis and 2 units up on the y-axis. This grid-like format allows us to easily visualize and calculate the distance between points, as well as other geometric properties like slope or midpoint.

Midpoint Formula

The midpoint formula is a way to find the exact center point of a line segment that connects two points on the coordinate plane. The formula for the midpoint, often denoted as \((M_x, M_y)\), is:\[(M_x, M_y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two endpoints of the line segment.

Applying this formula lets us find the midpoint exactly between the given points. By substituting the values for our points \((6, -2)\) and \((-6, 2)\),we compute: \[M_x = \frac{6 + (-6)}{2} = 0, \quad M_y = \frac{-2 + 2}{2} = 0\]So, the midpoint turns out to be (0, 0).This portion of the graph is balanced exactly between both points, representing the line segment's center.

Graphing Points

Graphing points accurately in a coordinate plane helps visualize problems and analyze spatial relationships. Here's how you can effectively graph points:Start by identifying each point's coordinates, given in pairs like \((x, y)\).

• Move horizontally along the x-axis to match the x-coordinate.
• Then, move vertically parallel to the y-axis to match the y-coordinate.

Each point marks a specific location on the plane. For example, when plotting the point \((6, -2)\), start at the origin, move 6 units right, and then 2 units down.For point \((-6, 2)\), begin at the origin, go 6 units left, then 2 units up.This process helps translate numerical coordinates into a visual layout, enabling the measurement of distances or finding midpoints.Visualizing these points aids not only in understanding the relationship between them but also provides insights into solving complex geometric problems involving them.