Problem 11 - Algebra and Trigonometry

Question

\(7-14\) 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. $$ (6,-2),(-1,3) $$

Step-by-Step Solution

Answer

The points are (6,-2) and (-1,3). Distance: \(\sqrt{74}\); Midpoint: \((\frac{5}{2}, \frac{1}{2})\).

Step 1: Plot the Points

Start by plotting the given points on a coordinate plane. The first point is \((6, -2)\) and the second point is \((-1, 3)\). Use the horizontal axis for the x-coordinates and the vertical axis for the y-coordinates. Mark and label the points accordingly.

Step 2: Calculate the Distance between the Points

Use the distance formula to find the distance between the points \((6, -2)\) and \((-1, 3)\). The distance formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the given points into the formula:\[ d = \sqrt{(-1 - 6)^2 + (3 - (-2))^2} \]\[ d = \sqrt{(-7)^2 + (5)^2} \]\[ d = \sqrt{49 + 25} \]\[ d = \sqrt{74} \]Thus, the distance between the points is \(\sqrt{74}\) or approximately \(8.6\).

Step 3: Find the Midpoint of the Segment

To find the midpoint of the segment joining the points \((6, -2)\) and \((-1, 3)\), use the midpoint formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Substitute the given values into the formula:\[ M = \left( \frac{6 + (-1)}{2}, \frac{-2 + 3}{2} \right) \]\[ M = \left( \frac{5}{2}, \frac{1}{2} \right) \]So, the midpoint of the segment is \(\left( \frac{5}{2}, \frac{1}{2} \right)\).

Key Concepts

coordinate plane

The coordinate plane is a two-dimensional space determined by two axes: the horizontal axis, known as the x-axis, and the vertical axis, known as the y-axis. These axes intersect at a point called the origin. The origin is the center of the coordinate system and is denoted as \(0, 0\).In this system, every point is represented by a pair of numbers \(x, y\), which are known as coordinates. The first number corresponds to the position on the x-axis, and the second number corresponds to the position on the y-axis. When working with a coordinate plane:

• The right side of the x-axis represents positive values while the left side depicts negative values.
• Above the origin on the y-axis are positive values, and below are negative values.

This plane is crucial for plotting points, graphing equations, and visualizing data in mathematics and other analytical fields.

midpoint formula

The midpoint formula is a handy tool in geometry to identify the center point of a line segment on a coordinate plane. Imagine you want to find the exact middle point between two given points. The midpoint formula enables this calculation using simple geometry.The formula itself is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Here, \(x_1\) and \(x_2\) are the x-coordinates of the two points, and \(y_1\) and \(y_2\) are the y-coordinates. By computing the average of the x-coordinates and y-coordinates separately, the formula provides the midpoint coordinates.Let's say we have the points \(6, -2\) and \(-1, 3\). Using the midpoint formula:1. Calculate the average of the x-coordinates: \[ \frac{6 + (-1)}{2} = \frac{5}{2} \]2. Calculate the average of the y-coordinates: \[ \frac{-2 + 3}{2} = \frac{1}{2} \]Putting these together, the midpoint of the segment is at \left( \frac{5}{2}, \frac{1}{2} \right)\. Finding this midpoint can be quite useful in various applications, such as dividing a section into equal parts or determining symmetry.

plotting points

Plotting points on a coordinate plane is an essential skill in mathematics. It involves accurately placing a point on the graph according to its coordinates. Let's break down the steps for plotting with an easy guide.First, take the coordinate pair \(x, y\). The x-coordinate indicates how far to move along the horizontal axis. If the x-coordinate is positive, move right; if it's negative, move left. Next, consider the y-coordinate. This figure tells you how far to move along the vertical axis. If the y-coordinate is positive, go up; if negative, go down.For example, to plot the point \(6, -2\):

• Begin at the origin \(0, 0\).
• Move 6 units to the right since the x-coordinate is 6.
• Then, move 2 units downward because the y-coordinate is -2.
• Mark the point on your graph.

Repeat these steps for any additional points you need to plot. Accurate plotting is foundational for creating graphs and solving geometric problems, making this a key competency in math and science disciplines.