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Magazine Chapter 1: Problem 2
In Problems \(1-4\), plot the given points in the coordinate plane and then find the distance between them. \((-3,5),(2,-2)\)
Short Answer
Expert verified
The distance between the points \((-3,5)\) and \((2,-2)\) is \(\sqrt{74}\).
Step by step solution
01
Understand the Coordinates
The problem provides two points, \((-3,5)\) and \((2,-2)\). These points are in the form \((x_1, y_1)\) and \((x_2, y_2)\) where \(x_1 = -3\), \(y_1 = 5\), \(x_2 = 2\), and \(y_2 = -2\). In order to plot the points, we identify the x-coordinate as the horizontal placement and the y-coordinate as the vertical placement.
02
Plot the Points
Draw a coordinate plane with horizontal and vertical axes. Locate the first point at the position where x is -3 and y is 5, and mark it. Similarly, locate the second point at x = 2 and y = -2 and mark it. These marks represent the points \((-3,5)\) and \((2,-2)\) on the graph.
03
Apply the Distance Formula
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is found using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the given values into the formula:\[d = \sqrt{(2 - (-3))^2 + (-2 - 5)^2}\]
04
Calculate the Difference for Each Coordinate
Find the difference in the x-coordinates: \(x_2 - x_1 = 2 - (-3) = 2 + 3 = 5\). Find the difference in the y-coordinates: \(y_2 - y_1 = -2 - 5 = -7\).
05
Find the Squares of the Differences
Calculate the square of each difference:\((x_2 - x_1)^2 = 5^2 = 25\)\((y_2 - y_1)^2 = (-7)^2 = 49\)
06
Find the Sum of the Squares
Add the squares of the differences:\(25 + 49 = 74\).
07
Calculate the Distance
Take the square root of the sum of the squares to find the distance:\(d = \sqrt{74}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distance Formula
The distance formula is a key concept in coordinate geometry. It helps us find the length of a line segment between two points on the coordinate plane. In general, the formula is given by:
- \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points defined by two numbers: an x-coordinate and a y-coordinate. It is initiated by two perpendicular lines called axes:
- The horizontal line is the x-axis.
- The vertical line is the y-axis.
Plotting Points
Plotting points involves locating and marking specific positions on the coordinate plane based on given coordinates. Here鈥檚 how you can plot a point efficiently:
- Start by identifying the x-coordinate, which tells you how far to move horizontally from the origin. A positive x moves right, whereas a negative x moves left.
- Next, identify the y-coordinate, indicating your vertical movement from the origin: upwards if positive, downwards if negative.
- For example, for the point (-3, 5), you first move 3 units left from the origin on the x-axis, then 5 units up on the y-axis.
- Similarly, for the point (2, -2), move 2 units right and then 2 units down.
Calculating Distance
Calculating distance using coordinates involves several steps that require careful arithmetic to ensure accuracy. Here鈥檚 a simple breakdown of the process:
- First, identify the differences in the corresponding coordinates of the two points: calculate \(x_2 - x_1\) and \(y_2 - y_1\).
- Then, square each of these differences to eliminate any negative signs and work with positive values: \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\).
- Add the squared values to determine the sum of the squares: this represents the square of the distance.
- Finally, take the square root of the sum you calculated. This step finds the actual distance, giving you a non-negative number that represents the shortest path between the two points.
