Problem 10
Question
find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-2) \text { and }(4,3) $$
Step-by-Step Solution
Verified Answer
The distance between the points (0,-2) and (4,3) is approximately 6.40.
1Step 1: Identify the coordinates
The given points are (0,-2) and (4,3). Here, \(x1 = 0\), \(y1 = -2\), \(x2 = 4\), and \(y2 = 3\).
2Step 2: Substitute in the distance formula
The distance d between two points \((x1, y1)\) and \((x2, y2)\) is \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\). Substituting the identified coordinates into the formula gives us \(d = \sqrt{(4 - 0)^2 + (3 - (-2))^2}\).
3Step 3: Simplify the equation
Simplify inside the square root: \(d = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41}\).
4Step 4: Calculate the square root
The square root of 41 is approximately 6.40, assuming that we are rounding to two decimal places.
Key Concepts
Understanding Coordinate GeometryFinding the Distance Between Two PointsApplying the Pythagorean Theorem
Understanding Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a crucial branch in the field of mathematics that allows us to precisely locate points, lines, and shapes in a two-dimensional space using numbers and equations. It represents geometric figures in a coordinate plane, which is made up of two number lines that intersect at a right angle forming what we call the origin.
The two axes, commonly referred to as the x-axis (horizontal) and the y-axis (vertical), enable us to define points with pairs of numbers called coordinates—the first number being the x-coordinate and the second the y-value. For instance, the coordinates \(0,-2\) represent a point that is located on the origin of the x-axis and 2 units below the origin on the y-axis.
To better understand how we can navigate this plane, imagine it as a map where every location has a specific address which is given as a pair of numbers—these numbers tell you how far along and how far up you need to travel from the origin to reach the desired point. By using these coordinates, we can perform various operations such as calculating distance, midpoint, slope, and area, thus enabling us to solve many practical and theoretical problems.
The two axes, commonly referred to as the x-axis (horizontal) and the y-axis (vertical), enable us to define points with pairs of numbers called coordinates—the first number being the x-coordinate and the second the y-value. For instance, the coordinates \(0,-2\) represent a point that is located on the origin of the x-axis and 2 units below the origin on the y-axis.
To better understand how we can navigate this plane, imagine it as a map where every location has a specific address which is given as a pair of numbers—these numbers tell you how far along and how far up you need to travel from the origin to reach the desired point. By using these coordinates, we can perform various operations such as calculating distance, midpoint, slope, and area, thus enabling us to solve many practical and theoretical problems.
Finding the Distance Between Two Points
The distance between two points in a coordinate plane can be determined using the distance formula, which is derived from the Pythagorean theorem. This formula is incredibly useful because it allows us to calculate the distance as if we were measuring a straight line between two points on a map.
To find the distance, you need the coordinates of the two points. As described in the solution, if you have two points, \(x1, y1\) and \(x2, y2\), the distance \(d\) between these points is calculated by the formula \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\).
This essentially treats the line between the two points as the hypotenuse of a right triangle, and the differences in x and y coordinates as the two other sides. In the example given, the coordinates (0,-2) and (4,3) represent two points. Substituting these into the distance formula gives us \(d = \sqrt{(4 - 0)^2 + (3 - (-2))^2}\), which simplifies down to \(d = \sqrt{41} \approx 6.40\) when rounded to two decimal places. It's a neat way to measure 'as the crow flies' distance in a flat plane.
To find the distance, you need the coordinates of the two points. As described in the solution, if you have two points, \(x1, y1\) and \(x2, y2\), the distance \(d\) between these points is calculated by the formula \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\).
This essentially treats the line between the two points as the hypotenuse of a right triangle, and the differences in x and y coordinates as the two other sides. In the example given, the coordinates (0,-2) and (4,3) represent two points. Substituting these into the distance formula gives us \(d = \sqrt{(4 - 0)^2 + (3 - (-2))^2}\), which simplifies down to \(d = \sqrt{41} \approx 6.40\) when rounded to two decimal places. It's a neat way to measure 'as the crow flies' distance in a flat plane.
Applying the Pythagorean Theorem
The Pythagorean theorem is a foundational principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
The theorem can be expressed in the simple equation: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs of the triangle, and \(c\) is the length of the hypotenuse. When we apply this theorem to coordinate geometry for finding the distance between two points, we are actually constructing an invisible right-angled triangle between those points and using the horizontal and the vertical distances between them as the legs \(a\) and \(b\).
In the context of the example we're examining, the distance calculation uses the differences in x and y coordinates as the lengths of the legs to find the length of the hypotenuse, which corresponds to the distance between the two points. Thus, the Pythagorean theorem underpins the logic of the distance formula in coordinate geometry, showing the beautiful interconnection between algebra and geometry.
The theorem can be expressed in the simple equation: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs of the triangle, and \(c\) is the length of the hypotenuse. When we apply this theorem to coordinate geometry for finding the distance between two points, we are actually constructing an invisible right-angled triangle between those points and using the horizontal and the vertical distances between them as the legs \(a\) and \(b\).
In the context of the example we're examining, the distance calculation uses the differences in x and y coordinates as the lengths of the legs to find the length of the hypotenuse, which corresponds to the distance between the two points. Thus, the Pythagorean theorem underpins the logic of the distance formula in coordinate geometry, showing the beautiful interconnection between algebra and geometry.
Other exercises in this chapter
Problem 9
Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises fal
View solution Problem 9
Determine whether each relation is a function. Give the domain and range for each relation. $$ \\{(1,4),(1,5),(1,6)\\} $$
View solution Problem 10
Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\sqrt[3]{x-4} \text { and } g(x
View solution Problem 10
Find the domain of each function. $$f(x)=\frac{1}{x+8}+\frac{3}{x-10}$$
View solution