Find The Distance Between The Points \[$(1, -3)\$\] And \[$(6, -8)\$\].If Necessary, Round Your Answer To The Nearest Tenth. \[$\square\$\] Units
Introduction
In mathematics, the distance between two points in a coordinate plane is a fundamental concept that has numerous applications in various fields, including geometry, trigonometry, and physics. The distance between two points can be calculated using the distance formula, which is a powerful tool for solving problems involving coordinate geometry. In this article, we will explore the concept of distance between two points in a coordinate plane and provide a step-by-step guide on how to calculate it.
What is the Distance Formula?
The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:
d = √((x2 - x1)² + (y2 - y1)²)
where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points, and √ represents the square root.
Step-by-Step Guide to Calculating the Distance
To calculate the distance between two points in a coordinate plane, follow these steps:
Step 1: Identify the Coordinates of the Two Points
The first step is to identify the coordinates of the two points. In this case, the coordinates of the two points are (1, -3) and (6, -8).
Step 2: Plug the Coordinates into the Distance Formula
Once you have identified the coordinates of the two points, plug them into the distance formula:
d = √((6 - 1)² + (-8 - (-3))²)
Step 3: Simplify the Expression
Simplify the expression by evaluating the squares and the subtraction:
d = √((5)² + (-5)²)
d = √(25 + 25)
d = √50
Step 4: Calculate the Square Root
Calculate the square root of 50:
d = √50 ≈ 7.071
Step 5: Round the Answer to the Nearest Tenth (If Necessary)
If necessary, round the answer to the nearest tenth. In this case, the answer is already rounded to the nearest tenth.
Conclusion
In conclusion, the distance between two points in a coordinate plane can be calculated using the distance formula. By following the steps outlined in this article, you can calculate the distance between two points in a coordinate plane. Remember to identify the coordinates of the two points, plug them into the distance formula, simplify the expression, calculate the square root, and round the answer to the nearest tenth (if necessary).
Example Problems
Here are some example problems to help you practice calculating the distance between two points in a coordinate plane:
- Find the distance between the points (2, 4) and (6, 8).
- Find the distance between the points (-3, 2) and (1, -4).
- Find the distance between the points (0, 0) and (3, 4).
Tips and Tricks
Here are some tips and tricks to help you calculate the distance between two points in a coordinate plane:
- Make sure to identify the coordinates of the two points correctly.
- Plug the coordinates into the distance formula correctly.
- Simplify the expression by evaluating the squares and the subtraction.
- Calculate the square root correctly.
- Round the answer to the nearest tenth (if necessary).
Common Mistakes
Here are some common mistakes to avoid when calculating the distance between two points in a coordinate plane:
- Failing to identify the coordinates of the two points correctly.
- Plugging the coordinates into the distance formula incorrectly.
- Failing to simplify the expression by evaluating the squares and the subtraction.
- Calculating the square root incorrectly.
- Failing to round the answer to the nearest tenth (if necessary).
Real-World Applications
The distance formula has numerous real-world applications in various fields, including:
- Navigation: The distance formula is used in navigation to calculate the distance between two points on a map.
- Physics: The distance formula is used in physics to calculate the distance between two objects in a coordinate plane.
- Engineering: The distance formula is used in engineering to calculate the distance between two points in a coordinate plane.
- Computer Science: The distance formula is used in computer science to calculate the distance between two points in a coordinate plane.
Conclusion
Frequently Asked Questions
In this article, we will answer some frequently asked questions about finding the distance between two points in a coordinate plane.
Q: What is the distance formula?
A: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:
d = √((x2 - x1)² + (y2 - y1)²)
where d is the distance between the two points, (x1, y1) and (x2, y2) are the coordinates of the two points, and √ represents the square root.
Q: How do I calculate the distance between two points in a coordinate plane?
A: To calculate the distance between two points in a coordinate plane, follow these steps:
- Identify the coordinates of the two points.
- Plug the coordinates into the distance formula.
- Simplify the expression by evaluating the squares and the subtraction.
- Calculate the square root.
- Round the answer to the nearest tenth (if necessary).
Q: What are some common mistakes to avoid when calculating the distance between two points in a coordinate plane?
A: Some common mistakes to avoid when calculating the distance between two points in a coordinate plane include:
- Failing to identify the coordinates of the two points correctly.
- Plugging the coordinates into the distance formula incorrectly.
- Failing to simplify the expression by evaluating the squares and the subtraction.
- Calculating the square root incorrectly.
- Failing to round the answer to the nearest tenth (if necessary).
Q: What are some real-world applications of the distance formula?
A: The distance formula has numerous real-world applications in various fields, including:
- Navigation: The distance formula is used in navigation to calculate the distance between two points on a map.
- Physics: The distance formula is used in physics to calculate the distance between two objects in a coordinate plane.
- Engineering: The distance formula is used in engineering to calculate the distance between two points in a coordinate plane.
- Computer Science: The distance formula is used in computer science to calculate the distance between two points in a coordinate plane.
Q: How do I round my answer to the nearest tenth?
A: To round your answer to the nearest tenth, follow these steps:
- Look at the hundredth place digit (the second digit after the decimal point).
- If the hundredth place digit is 5 or greater, round up the tenth place digit.
- If the hundredth place digit is less than 5, round down the tenth place digit.
Q: What if I have a negative distance?
A: If you have a negative distance, it means that the two points are on opposite sides of the coordinate plane. In this case, you can ignore the negative sign and calculate the distance as usual.
Q: Can I use the distance formula to calculate the distance between three points?
A: No, the distance formula is used to calculate the distance between two points in a coordinate plane. If you want to calculate the distance between three points, you will need to use a different formula or method.
Q: How do I calculate the distance between two points in a 3D coordinate plane?
A: To calculate the distance between two points in a 3D coordinate plane, you will need to use the distance formula in three dimensions. The formula is given by:
d = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)
where d is the distance between the two points, (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points, and √ represents the square root.
Conclusion
In conclusion, the distance formula is a powerful tool for calculating the distance between two points in a coordinate plane. By following the steps outlined in this article, you can calculate the distance between two points in a coordinate plane. Remember to identify the coordinates of the two points, plug them into the distance formula, simplify the expression, calculate the square root, and round the answer to the nearest tenth (if necessary).