MTH120 College Algebra Chapter 2.1

 2.1 The Rectangular Coordinate Systems and Graphs:

Plotting ordered pairs in the Cartesian coordinate system is a fundamental skill in mathematics and is often used to visualize data, functions, or relationships between variables. The Cartesian coordinate system, also known as the rectangular coordinate system, consists of two perpendicular axes: the x-axis and the y-axis. Here's how to plot ordered pairs on this system:

Step 1: Understand the Axes:

• The horizontal axis is called the x-axis, and it runs horizontally from left to right.
• The vertical axis is called the y-axis, and it runs vertically from bottom to top.
• The point where the x-axis and y-axis intersect is called the origin and is denoted as (0, 0).

Step 2: Identify the Ordered Pair:

• An ordered pair consists of two numbers (x, y), where x represents the horizontal position (left or right) and y represents the vertical position (up or down) in relation to the axes.

Step 3: Locate the Point:

• To plot the ordered pair (x, y), start at the origin (0, 0).
• Move horizontally (left or right) along the x-axis to the value of x.
• Move vertically (up or down) along the y-axis to the value of y.
• The point where you end up is the plotted point (x, y).

Step 4: Label the Point (Optional):

• You can label the plotted point with its coordinates, which helps in identifying the location of the point.

Step 5: Repeat for Additional Points:

• If you have more ordered pairs, repeat the process for each one, plotting them one by one.

Let's illustrate this with an example:

Example: Plot the ordered pairs (2, 3), (-1, 4), and (0, -2) on the Cartesian coordinate system.

Step 1: Understand the Axes:

• We have the x-axis (horizontal) and the y-axis (vertical).

Step 2: Identify the Ordered Pair:

• We have three ordered pairs: (2, 3), (-1, 4), and (0, -2).

Step 3: Locate the Points:

• For (2, 3), start at the origin (0, 0), move 2 units to the right along the x-axis, and then move 3 units up along the y-axis. You'll end up at the point (2, 3).
• For (-1, 4), start at the origin (0, 0), move 1 unit to the left along the x-axis, and then move 4 units up along the y-axis. You'll end up at the point (-1, 4).
• For (0, -2), start at the origin (0, 0), and move 2 units down along the y-axis. You'll end up at the point (0, -2).

Step 4: Label the Points (Optional):

• You can label each point with its coordinates if you wish.

Step 5: Repeat for Additional Points:

• In this example, we've plotted all three ordered pairs.

Now you have successfully plotted the ordered pairs (2, 3), (-1, 4), and (0, -2) on the Cartesian coordinate system. You can use this method to visualize and analyze various mathematical relationships and data points.

The Cartesian coordinate system, also known as the rectangular coordinate system, is a fundamental mathematical tool used for representing points and graphing functions in two-dimensional space. It was developed by the French mathematician and philosopher René Descartes in the 17th century and is named after him.

The Cartesian coordinate system consists of two perpendicular axes:

1. X-Axis: The horizontal axis is labeled as the x-axis. It runs horizontally from left to right.

2. Y-Axis: The vertical axis is labeled as the y-axis. It runs vertically from bottom to top.

The point where the x-axis and y-axis intersect is called the origin and is denoted as (0, 0). The origin serves as the reference point for all other points in the coordinate system.

To specify a point in the Cartesian coordinate system, you use an ordered pair of numbers (x, y), where:

• "x" represents the horizontal position of the point relative to the origin (left or right).
• "y" represents the vertical position of the point relative to the origin (up or down).

Here's how to plot and locate points in the Cartesian coordinate system:

1. Plotting Points: To plot a point (x, y), start at the origin (0, 0), move along the x-axis to the right if x is positive or to the left if x is negative, and then move along the y-axis upward if y is positive or downward if y is negative. The point where you end up is the plotted point (x, y).

2. Quadrants: The Cartesian plane is divided into four quadrants:

   • Quadrant I: Contains points with positive x and positive y coordinates.
   • Quadrant II: Contains points with negative x and positive y coordinates.
   • Quadrant III: Contains points with negative x and negative y coordinates.
   • Quadrant IV: Contains points with positive x and negative y coordinates.

3. Axes: The x-axis and y-axis divide the coordinate plane into four equal sections, and they are perpendicular to each other.

4. Distance: You can calculate the distance between two points in the Cartesian coordinate system using the distance formula, which is derived from the Pythagorean theorem.

The Cartesian coordinate system is a powerful tool used in various fields of mathematics, science, engineering, and more. It allows for the representation of data, functions, and geometric shapes, making it a fundamental concept in mathematics and a basis for understanding analytical geometry and algebra.

Graphing equations by plotting points is a common method used to visualize functions and relationships between variables. To graph an equation in this way, follow these steps:

Step 1: Choose a Range of Values for One Variable:

• Select a range of values for one of the variables, typically x. This will determine the x-coordinates of the points you'll plot on the graph.

Step 2: Calculate the Corresponding Values for the Other Variable:

• Use the given equation to calculate the corresponding values for the other variable, typically y. Substitute each x-value from the chosen range into the equation to find the corresponding y-values.

Step 3: Create Ordered Pairs:

• Pair each x-value with its corresponding y-value to create ordered pairs (x, y).

Step 4: Plot the Points:

• On a graph paper or a coordinate plane, locate the x and y axes. Label them appropriately.
• For each ordered pair, plot a point on the graph by moving horizontally along the x-axis to the x-coordinate and then moving vertically to the y-coordinate.

Step 5: Connect the Points:

• Once you've plotted several points, connect them with a smooth curve or line. This represents the graph of the equation.

Step 6: Label the Graph (Optional):

• Add labels and scales to the axes as needed, and provide any necessary information about the graph, such as units or key points.

Let's work through an example to illustrate this process:

Example: Graph the equation $�=2�-1$ by plotting points.

Step 1: Choose a Range of Values for x:

• Let's choose a range of x-values, for example, -2, -1, 0, 1, and 2.

Step 2: Calculate the Corresponding Values for y:

• Substitute each x-value into the equation to find the corresponding y-values:
  • For x = -2: $�=2(-2)-1=-4-1=-5$
  • For x = -1: $�=2(-1)-1=-2-1=-3$
  • For x = 0: $�=2(0)-1=0-1=-1$
  • For x = 1: $�=2(1)-1=2-1=1$
  • For x = 2: $�=2(2)-1=4-1=3$

Step 3: Create Ordered Pairs:

• Pair each x-value with its corresponding y-value:
  • (-2, -5), (-1, -3), (0, -1), (1, 1), (2, 3)

Step 4: Plot the Points:

• On a graph or coordinate plane, plot the points corresponding to the ordered pairs:

  • (-2, -5)
  • (-1, -3)
  • (0, -1)
  • (1, 1)
  • (2, 3)

Step 5: Connect the Points:

• Connect the plotted points with a straight line. This line represents the graph of the equation $�=2�-1$.

Step 6: Label the Graph (Optional):

• Label the x and y axes, add scales, and provide any necessary information about the graph.

You've successfully graphed the equation $�=2�-1$ by plotting points on a graph. This method allows you to visualize the relationship between x and y as a straight line in this case.

Let's work through two examples of graphing equations in two variables by plotting points.

Example 1: Graph the equation $�={�}^2$ by plotting points.

Step 1: Choose a Range of Values for x:

• Select a range of x-values. Let's choose -2, -1, 0, 1, and 2.

Step 2: Calculate the Corresponding Values for y:

• Use the equation $�={�}^2$ to calculate the corresponding y-values for each x-value:
  • For x = -2: $�=(-2{)}^2=4$
  • For x = -1: $�=(-1{)}^2=1$
  • For x = 0: $�=(0{)}^2=0$
  • For x = 1: $�=(1{)}^2=1$
  • For x = 2: $�=(2{)}^2=4$

Step 3: Create Ordered Pairs:

• Pair each x-value with its corresponding y-value:
  • (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)

Step 4: Plot the Points:

• On a graph or coordinate plane, plot the points corresponding to the ordered pairs:

  • (-2, 4)
  • (-1, 1)
  • (0, 0)
  • (1, 1)
  • (2, 4)

Step 5: Connect the Points:

• Connect the plotted points smoothly. This represents the graph of the equation $�={�}^2$.

Step 6: Label the Graph (Optional):

• Label the x and y axes, add scales, and provide any necessary information about the graph.

The graph of the equation $�={�}^2$ is a parabola that opens upward and passes through the points you plotted.

Example 2: Graph the equation $�=-2�+3$ by plotting points.

Step 1: Choose a Range of Values for x:

• Select a range of x-values. Let's choose -2, -1, 0, 1, and 2.

Step 2: Calculate the Corresponding Values for y:

• Use the equation $�=-2�+3$ to calculate the corresponding y-values for each x-value:
  • For x = -2: $�=-2(-2)+3=4+3=7$
  • For x = -1: $�=-2(-1)+3=2+3=5$
  • For x = 0: $�=-2(0)+3=0+3=3$
  • For x = 1: $�=-2(1)+3=-2+3=1$
  • For x = 2: $�=-2(2)+3=-4+3=-1$

Step 3: Create Ordered Pairs:

• Pair each x-value with its corresponding y-value:
  • (-2, 7), (-1, 5), (0, 3), (1, 1), (2, -1)

Step 4: Plot the Points:

• On a graph or coordinate plane, plot the points corresponding to the ordered pairs:

  • (-2, 7)
  • (-1, 5)
  • (0, 3)
  • (1, 1)
  • (2, -1)

Step 5: Connect the Points:

• Connect the plotted points smoothly. This represents the graph of the equation $�=-2�+3$.

Step 6: Label the Graph (Optional):

• Label the x and y axes, add scales, and provide any necessary information about the graph.

The graph of the equation $�=-2�+3$ is a straight line with a slope of -2 and a y-intercept of 3. It passes through the points you plotted.

Graphing equations with a graphing utility, such as a graphing calculator or graphing software, is a convenient way to visualize functions and relationships between variables. These tools are especially useful for graphing complex or transcendental functions that may be difficult to plot manually. Here are the general steps for graphing equations using a graphing utility:

Step 1: Turn on the Graphing Utility:

• Turn on your graphing calculator or launch your graphing software.

Step 2: Input the Equation:

• Enter the equation you want to graph into the graphing utility. Typically, there's an input or equation entry screen where you can type or input mathematical expressions.

Step 3: Set the Graphing Window:

• Define the range for the x-values (the x-axis) and the y-values (the y-axis). This is done to specify the viewing window for the graph.
• Determine the x-min, x-max, y-min, and y-max values to display the portion of the graph that you're interested in. You may need to consult the user manual or help menu for your specific graphing utility to learn how to set these values.

Step 4: Graph the Equation:

• Once you've entered the equation and set the graphing window, instruct the calculator or software to graph the equation.
• In many graphing utilities, this is done by pressing a button labeled "GRAPH" or a similar command.

Step 5: Analyze the Graph:

• Examine the graph displayed on the screen. You can use the navigation tools or controls to zoom in or out, pan, or trace the graph to gather more information.

Step 6: Customize the Graph (Optional):

• If needed, you can customize the appearance of the graph, such as adjusting the color, line style, or adding labels to axes.

Step 7: Save or Export the Graph (Optional):

• Some graphing utilities allow you to save or export the graph as an image or data file for use in reports or presentations.

Step 8: Interpret the Results:

• Analyze the graph to understand the behavior of the function. Identify key features such as intercepts, maxima, minima, asymptotes, and inflection points.

Graphing utilities provide a visual representation of mathematical functions and are valuable tools for exploring functions, solving equations, and gaining insights into mathematical relationships. They are commonly used in education and various fields of science and engineering. Remember to consult the user manual or documentation specific to your graphing utility for detailed instructions on its use.

Using a graphing utility to graph an equation is a straightforward process. Below, I'll provide step-by-step instructions using a general graphing calculator as an example. Keep in mind that specific models of calculators and graphing software may have slight variations in their interface, but the basic steps remain similar:

Step 1: Turn on the Graphing Calculator:

• Ensure your graphing calculator is powered on.

Step 2: Access the Equation Entry Screen:

• Most graphing calculators have an "Y=" button or similar label. Press this button to access the equation entry screen.

Step 3: Enter the Equation:

• Use the calculator's keypad to type in the equation you want to graph. Make sure you correctly input all terms, operators, and variables.
• Example: To graph $�=2{�}^2-3�+1$, you would enter "2x^2 - 3x + 1" on the equation entry screen.

Step 4: Define the Graphing Window:

• Set the range for the x-values and y-values that you want to view on the graph. This step defines the window for your graph.
• Press the "WINDOW" button or a similar labeled button to access the window settings.
• Adjust the following settings as needed:
  • Xmin: The minimum x-value.
  • Xmax: The maximum x-value.
  • Ymin: The minimum y-value.
  • Ymax: The maximum y-value.
• Make sure these values encompass the portion of the graph you're interested in. You may also specify the scale for the axes (XTick and YTick) and other settings.

Step 5: Graph the Equation:

• After you've entered the equation and defined the graphing window, press the "GRAPH" or "DRAW" button (label may vary by calculator model). This will instruct the calculator to plot the graph of the equation.

Step 6: Interpret the Graph:

• Examine the graph displayed on the screen. You can use the navigation buttons or commands (e.g., zoom in, zoom out) to explore the graph further.
• Identify key features of the graph, such as intercepts, maxima, minima, asymptotes, and any other relevant information.

Step 7: Customize the Graph (Optional):

• If needed, you can customize the appearance of the graph by adjusting settings related to line style, color, labels, and more.

Step 8: Save or Export the Graph (Optional):

• Some graphing calculators allow you to save or export the graph as an image or data file for use in reports or presentations.

Using a graphing calculator or graphing software is a valuable tool for visualizing mathematical functions and analyzing their behavior. Remember that different calculator models may have variations in their user interface, so consulting your calculator's manual for specific instructions is always a good practice.

To find the x-intercepts and y-intercepts of a function or equation, you're essentially looking for the points where the graph of the equation crosses the x-axis (x-intercepts) and the y-axis (y-intercepts). Here's how you can find them:

X-Intercepts (Roots or Zeros):

1. Set $�=0$: To find the x-intercepts, set $�$ in your equation equal to 0.

2. Solve for $�$: Solve the equation for $�$. The solutions (values of $�$) are the x-intercepts.

3. Express as Ordered Pairs: The x-intercepts are typically expressed as ordered pairs in the form $(�,0)$, where $�$ is the value you found in step 2, and 0 represents the y-coordinate.

Y-Intercepts:

1. Set $�=0$: To find the y-intercepts, set $�$ in your equation equal to 0.

2. Solve for $�$: Solve the equation for $�$. The solution (value of $�$) is the y-intercept.

3. Express as an Ordered Pair: The y-intercept is typically expressed as an ordered pair in the form $(0,�)$, where $�$ is the value you found in step 2, and 0 represents the x-coordinate.

Here are some examples to illustrate the process:

Example 1: Find the x-intercepts and y-intercepts of the equation $�=2�-4$.

X-Intercepts:

1. Set $�=0$: $0=2�-4$

2. Solve for $�$: $2�=4$ $\Rightarrow$ $�=2$

3. Express as Ordered Pairs: The x-intercept is $(2,0)$.

Y-Intercepts:

1. Set $�=0$: $�=2(0)-4$

2. Solve for $�$: $�=-4$

3. Express as an Ordered Pair: The y-intercept is $(0,-4)$.

So, the x-intercept is $(2,0)$, and the y-intercept is $(0,-4)$.

Example 2: Find the x-intercepts and y-intercepts of the equation $�={�}^2-9$.

X-Intercepts:

1. Set $�=0$: $0={�}^2-9$

2. Solve for $�$: ${�}^2=9$

3. Take the square root of both sides: $�=\pm 3$

4. Express as Ordered Pairs: The x-intercepts are $(-3,0)$ and $(3,0)$.

Y-Intercepts:

1. Set $�=0$: $�=(0{)}^2-9$

2. Solve for $�$: $�=-9$

3. Express as an Ordered Pair: The y-intercept is $(0,-9)$.

So, the x-intercepts are $(-3,0)$ and $(3,0)$, and the y-intercept is $(0,-9)$.

By finding the x-intercepts and y-intercepts, you can better understand the behavior and key points of a function's graph.

The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. It is a direct application of the Pythagorean theorem and is commonly used in geometry and trigonometry. The formula is as follows:

Distance Formula: The distance ($�$) between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in a coordinate plane is given by:

$�=\sqrt{({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2}$

Here's how to use the distance formula step by step:

Step 1: Identify the Two Points:

• You have two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in the coordinate plane. These are the two points between which you want to find the distance.

Step 2: Plug the Coordinates into the Formula:

• Substitute the coordinates of the two points into the distance formula.
• $({�}_1,{�}_1)$ goes in place of $({�}_1,{�}_1)$, and $({�}_2,{�}_2)$ goes in place of $({�}_2,{�}_2)$.

Step 3: Calculate the Differences:

• Calculate the differences between the x-coordinates and the y-coordinates:
  • $({�}_2-{�}_1)$ and $({�}_2-{�}_1)$.

Step 4: Square the Differences:

• Square the differences calculated in step 3:
  • $({�}_2-{�}_1{)}^2$ and $({�}_2-{�}_1{)}^2$.

Step 5: Add the Squares:

• Add the squared differences together:
  • $({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2$.

Step 6: Take the Square Root:

• Take the square root of the sum from step 5 to find the distance:
  • $�=\sqrt{({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2}$.

This final value ($�$) is the distance between the two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in the coordinate plane.

Example: Calculate the distance between the points $(-1,2)$ and $(3,5)$.

Step 1: Identify the Two Points:

• $({�}_1,{�}_1)=(-1,2)$
• $({�}_2,{�}_2)=(3,5)$

Step 2: Plug the Coordinates into the Formula:

• Substitute the coordinates into the distance formula:
  • $�=\sqrt{(3-(-1){)}^2+(5-2{)}^2}$

Step 3: Calculate the Differences:

• Calculate the differences between the x-coordinates and the y-coordinates:
  • ${�}_2-{�}_1=3-(-1)=4$
  • ${�}_2-{�}_1=5-2=3$

Step 4: Square the Differences:

• Square the differences calculated in step 3:
  • $(4{)}^2=16$
  • $(3{)}^2=9$

Step 5: Add the Squares:

• Add the squared differences together:
  • $16+9=25$

Step 6: Take the Square Root:

• Take the square root of the sum from step 5:
  • $�=\sqrt{25}=5$

So, the distance between the points $(-1,2)$ and $(3,5)$ is 5 units.

Let's work through two examples of finding the distance between two points using the distance formula.

Example 1: Find the distance between the points A(2, 3) and B(6, 7).

Step 1: Identify the Two Points:

• Point A: $({�}_1,{�}_1)=(2,3)$
• Point B: $({�}_2,{�}_2)=(6,7)$

Step 2: Plug the Coordinates into the Formula:

• Substitute the coordinates into the distance formula:
  • $�=\sqrt{(6-2{)}^2+(7-3{)}^2}$

Step 3: Calculate the Differences:

• Calculate the differences between the x-coordinates and the y-coordinates:
  • ${�}_2-{�}_1=6-2=4$
  • ${�}_2-{�}_1=7-3=4$

Step 4: Square the Differences:

• Square the differences calculated in step 3:
  • $(4{)}^2=16$
  • $(4{)}^2=16$

Step 5: Add the Squares:

• Add the squared differences together:
  • $16+16=32$

Step 6: Take the Square Root:

• Take the square root of the sum from step 5:
  • $�=\sqrt{32}\approx 5.66$

So, the distance between the points A(2, 3) and B(6, 7) is approximately 5.66 units.

Example 2: Find the distance between the points C(-1, -2) and D(4, 5).

Step 1: Identify the Two Points:

• Point C: $({�}_1,{�}_1)=(-1,-2)$
• Point D: $({�}_2,{�}_2)=(4,5)$

Step 2: Plug the Coordinates into the Formula:

• Substitute the coordinates into the distance formula:
  • $�=\sqrt{(4-(-1){)}^2+(5-(-2){)}^2}$

Step 3: Calculate the Differences:

• Calculate the differences between the x-coordinates and the y-coordinates:
  • ${�}_2-{�}_1=4-(-1)=5$
  • ${�}_2-{�}_1=5-(-2)=7$

Step 4: Square the Differences:

• Square the differences calculated in step 3:
  • $(5{)}^2=25$
  • $(7{)}^2=49$

Step 5: Add the Squares:

• Add the squared differences together:
  • $25+49=74$

Step 6: Take the Square Root:

• Take the square root of the sum from step 5:
  • $�=\sqrt{74}\approx 8.60$

So, the distance between the points C(-1, -2) and D(4, 5) is approximately 8.60 units.

To find the distance between two locations on the Earth's surface (given their latitude and longitude coordinates), you can use the Haversine formula. The Haversine formula calculates the distance between two points on the surface of a sphere (like the Earth) based on their latitudes and longitudes. Here's how you can use the Haversine formula to find the distance:

Haversine Formula: The formula to calculate the distance ($�$) between two points $({\text{lat}}_1,{\text{lon}}_1)$ and $({\text{lat}}_2,{\text{lon}}_2)$ on the Earth's surface is:

$�=2�\arcsin \left(\sqrt{{\sin }^2\left(\frac{\mathrm{\Delta }\text{lat}}{2}\right)+\cos ({\text{lat}}_1)\cos ({\text{lat}}_2){\sin }^2\left(\frac{\mathrm{\Delta }\text{lon}}{2}\right)}\right)$

Where:

• $�$ is the distance between the two points.
• ${\text{lat}}_1$ and ${\text{lat}}_2$ are the latitudes of the two points in radians.
• ${\text{lon}}_1$ and ${\text{lon}}_2$ are the longitudes of the two points in radians.
• $\mathrm{\Delta }\text{lat}={\text{lat}}_2-{\text{lat}}_1$.
• $\mathrm{\Delta }\text{lon}={\text{lon}}_2-{\text{lon}}_1$.
• $�$ is the radius of the Earth. The mean radius of the Earth is approximately 6,371 kilometers (3,959 miles).

Here's a step-by-step process for finding the distance between two locations using the Haversine formula:

Step 1: Convert Degrees to Radians:

• Convert the latitude and longitude coordinates of the two locations from degrees to radians. You can do this by multiplying by $�\mathrm{/}180$.

Step 2: Calculate $\mathrm{\Delta }\text{lat}$ and $\mathrm{\Delta }\text{lon}$:

• Calculate the differences in latitude and longitude: $\mathrm{\Delta }\text{lat}={\text{lat}}_2-{\text{lat}}_1$ and $\mathrm{\Delta }\text{lon}={\text{lon}}_2-{\text{lon}}_1$.

Step 3: Apply the Haversine Formula:

• Use the Haversine formula to calculate the distance $�$ between the two points.

Step 4: Convert the Result:

• The result $�$ will be in the same units as the Earth's radius ($�$), which is typically kilometers or miles.

Let's work through an example:

Example: Find the distance between two locations with the following coordinates:

• Location A: Latitude 40.7128° N, Longitude 74.0060° W
• Location B: Latitude 34.0522° N, Longitude 118.2437° W

Step 1: Convert Degrees to Radians:

• ${\text{lat}}_1=40.7128\times (�\mathrm{/}180)\approx 0.7102$ radians
• ${\text{lon}}_1=(-74.0060)\times (�\mathrm{/}180)\approx -1.2923$ radians
• ${\text{lat}}_2=34.0522\times (�\mathrm{/}180)\approx 0.5946$ radians
• ${\text{lon}}_2=(-118.2437)\times (�\mathrm{/}180)\approx -2.0664$ radians

Step 2: Calculate $\mathrm{\Delta }\text{lat}$ and $\mathrm{\Delta }\text{lon}$:

• $\mathrm{\Delta }\text{lat}={\text{lat}}_2-{\text{lat}}_1\approx 0.5946-0.7102\approx -0.1156$ radians
• $\mathrm{\Delta }\text{lon}={\text{lon}}_2-{\text{lon}}_1\approx -2.0664-(-1.2923)\approx -0.7741$ radians

Step 3: Apply the Haversine Formula:

• Plug the values into the Haversine formula: $�=2\cdot 6371\cdot \arcsin \left(\sqrt{{\sin }^2\left(\frac{-0.1156}{2}\right)+\cos (0.7102)\cos (0.5946){\sin }^2\left(\frac{-0.7741}{2}\right)}\right)$

Step 4: Calculate the Distance:

• Perform the calculations to find the distance $�$. The result will be in kilometers (using the Earth's mean radius of 6,371 kilometers).
• $�\approx 4151$ kilometers

So, the distance between the two locations A and B is approximately 4,151 kilometers.

The midpoint formula allows you to find the coordinates of the midpoint between two given points in a coordinate plane. The formula is:

Midpoint Formula: The midpoint $({�}_{�},{�}_{�})$ between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ is given by:

${�}_{�}=\frac{{�}_1+{�}_2}{2}$ ${�}_{�}=\frac{{�}_1+{�}_2}{2}$

Here's how to use the midpoint formula step by step:

Step 1: Identify the Two Points:

• You have two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ between which you want to find the midpoint.

Step 2: Apply the Midpoint Formula:

• Use the midpoint formula to calculate the coordinates of the midpoint:
  • ${�}_{�}=\frac{{�}_1+{�}_2}{2}$
  • ${�}_{�}=\frac{{�}_1+{�}_2}{2}$

Step 3: Express the Midpoint:

• The midpoint is typically expressed as an ordered pair $({�}_{�},{�}_{�})$, where ${�}_{�}$ is the x-coordinate of the midpoint, and ${�}_{�}$ is the y-coordinate of the midpoint.

Here are some examples to illustrate the process:

Example 1: Find the midpoint between the points A(2, 4) and B(6, 8).

Step 1: Identify the Two Points:

• Point A: $({�}_1,{�}_1)=(2,4)$
• Point B: $({�}_2,{�}_2)=(6,8)$

Step 2: Apply the Midpoint Formula:

• Use the midpoint formula to calculate the coordinates of the midpoint:
  • ${�}_{�}=\frac{2+6}{2}=\frac{8}{2}=4$
  • ${�}_{�}=\frac{4+8}{2}=\frac{12}{2}=6$

Step 3: Express the Midpoint:

• The midpoint is $(4,6)$.

So, the midpoint between the points A(2, 4) and B(6, 8) is (4, 6).

Example 2: Find the midpoint between the points C(-3, 2) and D(5, -6).

Step 1: Identify the Two Points:

• Point C: $({�}_1,{�}_1)=(-3,2)$
• Point D: $({�}_2,{�}_2)=(5,-6)$

Step 2: Apply the Midpoint Formula:

• Use the midpoint formula to calculate the coordinates of the midpoint:
  • ${�}_{�}=\frac{-3+5}{2}=\frac{2}{2}=1$
  • ${�}_{�}=\frac{2-6}{2}=\frac{-4}{2}=-2$

Step 3: Express the Midpoint:

• The midpoint is $(1,-2)$.

So, the midpoint between the points C(-3, 2) and D(5, -6) is (1, -2).

To find the midpoint of a line segment with given endpoints, you can use the midpoint formula. The midpoint formula calculates the coordinates of the midpoint ($�$) between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ on a line segment and is as follows:

Midpoint Formula: The midpoint $�$ between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ is given by:

$�(\frac{{�}_1+{�}_2}{2},\frac{{�}_1+{�}_2}{2})$

Here's how to find the midpoint of a line segment step by step:

Step 1: Identify the Endpoints:

• You have two endpoints $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ that define the line segment.

Step 2: Apply the Midpoint Formula:

• Use the midpoint formula to calculate the coordinates of the midpoint:
  • ${�}_{�}=\frac{{�}_1+{�}_2}{2}$
  • ${�}_{�}=\frac{{�}_1+{�}_2}{2}$

Step 3: Express the Midpoint:

• The midpoint is typically expressed as an ordered pair $({�}_{�},{�}_{�})$, where ${�}_{�}$ is the x-coordinate of the midpoint, and ${�}_{�}$ is the y-coordinate of the midpoint.

Let's work through an example:

Example: Find the midpoint of the line segment with endpoints A(1, 2) and B(5, 8).

Step 1: Identify the Endpoints:

• Point A: $({�}_1,{�}_1)=(1,2)$
• Point B: $({�}_2,{�}_2)=(5,8)$

Step 2: Apply the Midpoint Formula:

• Use the midpoint formula to calculate the coordinates of the midpoint:
  • ${�}_{�}=\frac{1+5}{2}=\frac{6}{2}=3$
  • ${�}_{�}=\frac{2+8}{2}=\frac{10}{2}=5$

Step 3: Express the Midpoint:

• The midpoint is $(3,5)$.

So, the midpoint of the line segment with endpoints A(1, 2) and B(5, 8) is (3, 5). This means that the point (3, 5) is exactly halfway between A and B along the line segment.

To find the center of a circle, you'll need information about the circle's equation or its geometric properties. There are a few different scenarios:

1. Given the Equation of the Circle:

   • If you're given the equation of the circle in the form $(�-ℎ{)}^2+(�-�{)}^2={�}^2$, where $(ℎ,�)$ are the coordinates of the center and $�$ is the radius, then the center is at the point $(ℎ,�)$.

2. Given Three Points on the Circle:

   • If you're given three points that lie on the circle, you can use the method of finding the center of a circle that passes through three non-collinear points. Here are the steps:
     • Calculate the midpoints of two line segments formed by these three points.
     • Find the perpendicular bisectors of these two line segments.
     • The intersection point of the two perpendicular bisectors is the center of the circle.

3. Given Geometric Information:

   • If you have geometric information about the circle, such as its center and a point on its circumference, you can find the center using the midpoint formula.
     • The midpoint between the given point on the circumference and the center of the circle is the center.

Here's an example for each scenario:

Example 1: Given the equation of a circle $(�-2{)}^2+(�+3{)}^2=9$, find the center.

Solution:

• The equation is in the form $(�-ℎ{)}^2+(�-�{)}^2={�}^2$.
• The center is at the point $(ℎ,�)$, so in this case, the center is $(2,-3)$.

Example 2: Given three non-collinear points A(1, 2), B(4, 6), and C(7, 2) on the circumference of a circle, find the center of the circle.

Solution:

• Calculate the midpoints of two line segments:
  • Midpoint of AB: $(\frac{1+4}{2},\frac{2+6}{2})=(2.5,4)$
  • Midpoint of BC: $(\frac{4+7}{2},\frac{6+2}{2})=(5.5,4)$
• Find the perpendicular bisectors of AB and BC.
• The intersection point of these bisectors is the center of the circle, which is $(4,4)$.

Example 3: Given a circle with center C(3, -1) and a point on the circumference P(7, 3), find the center of the circle.

Solution:

• To find the center, you can use the midpoint formula.
• The midpoint between C(3, -1) and P(7, 3) is:
  • $(\frac{3+7}{2},\frac{-1+3}{2})=(5,1)$
• So, the center of the circle is (5, 1).

These examples illustrate how to find the center of a circle based on different types of information you may have about the circle.