Find The Exact Distance Between The Points \[$(13, 2)\$\] And \[$(-10, 0)\$\].
Introduction
In mathematics, particularly in geometry and trigonometry, calculating the distance between two points is a fundamental concept. This article will guide you through the process of finding the exact distance between two points in a coordinate plane. We will use the distance formula, which is a powerful tool for solving problems involving distance and coordinates.
What is the Distance Formula?
The distance formula is a mathematical formula used to find the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is widely used in various fields, including mathematics, physics, engineering, and computer science. The distance formula 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
Step-by-Step Guide to Finding the Distance
To find the distance between two points, follow these steps:
Step 1: Identify the Coordinates
The first step is to identify the coordinates of the two points. In this case, the coordinates are given as:
- (x1, y1) = (13, 2)
- (x2, y2) = (-10, 0)
Step 2: Plug in the Values
Next, plug in the values of the coordinates into the distance formula:
d = √((-10 - 13)² + (0 - 2)²)
Step 3: Simplify the Expression
Simplify the expression by evaluating the squares and the subtraction:
d = √((-23)² + (-2)²) d = √(529 + 4) d = √533
Step 4: Calculate the Square Root
Calculate the square root of 533 to find the exact distance:
d ≈ 23.07
Conclusion
In this article, we have learned how to use the distance formula to find the exact distance between two points in a coordinate plane. We have followed a step-by-step guide to simplify the expression and calculate the square root to find the final answer. The distance formula is a powerful tool that can be used to solve a wide range of problems involving distance and coordinates.
Real-World Applications
The distance formula has numerous real-world applications, including:
- Navigation: The distance formula is used in navigation systems to calculate the distance between two points on a map.
- Physics: The distance formula is used to calculate the distance traveled by an object under the influence of a constant acceleration.
- Computer Science: The distance formula is used in computer graphics to calculate the distance between two points in a 3D space.
Common Mistakes to Avoid
When using the distance formula, there are several common mistakes to avoid:
- Incorrectly identifying the coordinates: Make sure to identify the correct coordinates of the two points.
- Incorrectly plugging in the values: Make sure to plug in the correct values into the distance formula.
- Incorrectly simplifying the expression: Make sure to simplify the expression correctly to avoid errors.
Practice Problems
To practice using the distance formula, try solving the following problems:
- Find the distance between the points (3, 4) and (6, 8).
- Find the distance between the points (-2, 5) and (4, -3).
- Find the distance between the points (0, 0) and (3, 4).
Conclusion
Q: What is the distance formula?
A: The distance formula is a mathematical formula used to find 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
Q: How do I use the distance formula?
A: To use the distance formula, follow these steps:
- Identify the coordinates of the two points.
- Plug in the values of the coordinates into the distance formula.
- Simplify the expression by evaluating the squares and the subtraction.
- Calculate the square root to find the exact distance.
Q: What are some common mistakes to avoid when using the distance formula?
A: Some common mistakes to avoid when using the distance formula include:
- Incorrectly identifying the coordinates
- Incorrectly plugging in the values
- Incorrectly simplifying the expression
Q: Can I use the distance formula to find the distance between two points in 3D space?
A: Yes, you can use the distance formula to find the distance between two points in 3D space. However, you will need to use the 3D distance formula, which 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
Q: Can I use the distance formula to find the distance between two points on a map?
A: Yes, you can use the distance formula to find the distance between two points on a map. However, you will need to use the latitude and longitude coordinates of the two points instead of the x and y coordinates.
Q: Is the distance formula only used in mathematics?
A: No, the distance formula is used in a wide range of fields, including:
- Navigation
- Physics
- Computer Science
- Engineering
- Geography
Q: Can I use the distance formula to find the distance between two points on a sphere?
A: Yes, you can use the distance formula to find the distance between two points on a sphere. However, you will need to use the spherical distance formula, which is given by:
d = arccos(sin(φ1)sin(φ2) + cos(φ1)cos(φ2)cos(Δλ))
where:
- d is the distance between the two points
- φ1 and φ2 are the latitudes of the two points
- Δλ is the difference in longitude between the two points
Q: Is the distance formula a linear or non-linear formula?
A: The distance formula is a non-linear formula, as it involves the square root of a sum of squares.
Q: Can I use the distance formula to find the distance between two points in a non-Euclidean space?
A: No, the distance formula is only applicable in Euclidean space. In non-Euclidean space, you will need to use a different formula to find the distance between two points.
Conclusion
In conclusion, the distance formula is a powerful tool that can be used to find the distance between two points in a coordinate plane. By following a step-by-step guide and avoiding common mistakes, you can use the distance formula to solve a wide range of problems involving distance and coordinates.