Question 22 Of 42 What Is The Distance Between The Points (-4, 2) And (1, -3) On The Coordinate Plane? A. 7.07 Units B. 5.83 Units C. 7.21 Units D. 2.83 Units
The distance between two points on a coordinate plane can be calculated using the distance formula. This formula is derived from the Pythagorean theorem and is used to find the length of the line segment connecting two points in a two-dimensional space.
The Distance Formula
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Applying the Distance Formula
To find the distance between the points (-4, 2) and (1, -3), we can plug these values into the distance formula.
Step 1: Identify the coordinates
The coordinates of the two points are (-4, 2) and (1, -3).
Step 2: Plug the values into the distance formula
d = √((1 - (-4))^2 + (-3 - 2)^2) d = √((5)^2 + (-5)^2) d = √(25 + 25) d = √50
Step 3: Simplify the square root
To simplify the square root, we can find the prime factorization of 50.
50 = 2 × 5 × 5
Step 4: Simplify the square root
Now, we can simplify the square root by taking the square root of the factors.
d = √(2 × 5 × 5) d = √(2) × √(5 × 5) d = √(2) × 5 d = 5√(2)
Step 5: Approximate the value
To approximate the value of 5√(2), we can use a calculator or a table of square roots.
5√(2) ≈ 7.07
Conclusion
The distance between the points (-4, 2) and (1, -3) on the coordinate plane is approximately 7.07 units.
Answer
The correct answer is:
A. 7.07 units
Discussion
The distance formula is a powerful tool for finding the length of line segments in a two-dimensional space. By applying the distance formula, we can find the distance between any two points on the coordinate plane.
Example Problems
- Find the distance between the points (3, 4) and (6, 8).
- Find the distance between the points (-2, 1) and (4, -3).
- Find the distance between the points (0, 0) and (3, 4).
Practice Problems
- Find the distance between the points (2, 3) and (5, 7).
- Find the distance between the points (-1, 2) and (3, -4).
- Find the distance between the points (1, 1) and (4, 3).
Key Concepts
- The distance formula is used to find the length of line segments in a two-dimensional space.
- The distance formula is derived from the Pythagorean theorem.
- The distance formula is given by d = √((x2 - x1)^2 + (y2 - y1)^2).
- The distance formula can be used to find the distance between any two points on the coordinate plane.
Glossary
- Coordinate plane: A two-dimensional space with x and y axes.
- Distance formula: A formula used to find the length of line segments in a two-dimensional space.
- Pythagorean theorem: A theorem used to find the length of the hypotenuse of a right triangle.
- Square root: The inverse operation of squaring a number.
Q&A: Distance Formula ==========================
Q: What is the distance formula?
A: The distance formula is a mathematical formula used to find the length of a line segment in a two-dimensional space. It is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Q: How do I use the distance formula?
A: To use the distance formula, you need to know the coordinates of the two points. Plug the values into the formula and simplify the square root.
Q: What is the significance of the distance formula?
A: The distance formula is a powerful tool for finding the length of line segments in a two-dimensional space. It is used in various fields such as physics, engineering, and computer science.
Q: Can I use the distance formula to find the distance between any two points?
A: Yes, you can use the distance formula to find the distance between any two points on the coordinate plane.
Q: What is the difference between the distance formula and the Pythagorean theorem?
A: The distance formula is a specific application of the Pythagorean theorem. The Pythagorean theorem is a general theorem used to find the length of the hypotenuse of a right triangle, while the distance formula is used to find the distance between two points in a two-dimensional space.
Q: Can I use the distance formula to find the distance between three or more points?
A: No, the distance formula is used to find the distance between two points. To find the distance between three or more points, you need to use a different formula or method.
Q: How do I approximate the value of the square root?
A: You can use a calculator or a table of square roots to approximate the value of the square root.
Q: Can I use the distance formula to find the distance between points in a three-dimensional space?
A: No, the distance formula is used to find the distance between points in a two-dimensional space. To find the distance between points in a three-dimensional space, you need to use a different formula or method.
Q: What are some real-world applications of the distance formula?
A: The distance formula has many real-world applications, such as:
- Finding the distance between two cities or towns
- Calculating the distance between two points on a map
- Determining the length of a line segment in a two-dimensional space
- Finding the distance between two points in a computer graphics or game development context
Q: Can I use the distance formula to find the distance between points with negative coordinates?
A: Yes, you can use the distance formula to find the distance between points with negative coordinates.
Q: What is the difference between the distance formula and the Euclidean distance?
A: The distance formula and the Euclidean distance are the same thing. The Euclidean distance is a specific type of distance that is used to find the distance between two points in a two-dimensional space.
Q: Can I use the distance formula to find the distance between points in a non-Euclidean space?
A: No, the distance formula is used to find the distance between points in a Euclidean space. To find the distance between points in a non-Euclidean space, you need to use a different formula or method.
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:
- Not plugging in the correct values for the coordinates
- Not simplifying the square root correctly
- Not using the correct formula for the distance between two points in a two-dimensional space
Q: Can I use the distance formula to find the distance between points with decimal coordinates?
A: Yes, you can use the distance formula to find the distance between points with decimal coordinates.
Q: What is the relationship between the distance formula and the concept of distance?
A: The distance formula is a mathematical representation of the concept of distance. It is used to find the length of a line segment in a two-dimensional space.
Q: Can I use the distance formula to find the distance between points in a discrete space?
A: No, the distance formula is used to find the distance between points in a continuous space. To find the distance between points in a discrete space, you need to use a different formula or method.
Q: What are some advanced topics related to the distance formula?
A: Some advanced topics related to the distance formula include:
- The concept of distance in higher-dimensional spaces
- The use of the distance formula in machine learning and data analysis
- The application of the distance formula in computer graphics and game development
Q: Can I use the distance formula to find the distance between points in a non-metric space?
A: No, the distance formula is used to find the distance between points in a metric space. To find the distance between points in a non-metric space, you need to use a different formula or method.
Q: What is the relationship between the distance formula and the concept of similarity?
A: The distance formula is used to find the distance between points, which is related to the concept of similarity. The smaller the distance between two points, the more similar they are.
Q: Can I use the distance formula to find the distance between points in a space with non-standard coordinates?
A: No, the distance formula is used to find the distance between points in a space with standard coordinates. To find the distance between points in a space with non-standard coordinates, you need to use a different formula or method.
Q: What are some common applications of the distance formula in science and engineering?
A: Some common applications of the distance formula in science and engineering include:
- Calculating the distance between two objects in a physical system
- Determining the length of a line segment in a two-dimensional space
- Finding the distance between two points in a computer graphics or game development context
- Calculating the distance between two objects in a mechanical system
Q: Can I use the distance formula to find the distance between points in a space with non-integer coordinates?
A: Yes, you can use the distance formula to find the distance between points in a space with non-integer coordinates.
Q: What is the relationship between the distance formula and the concept of distance in physics?
A: The distance formula is used to find the distance between points, which is related to the concept of distance in physics. The distance formula is used to calculate the distance between two objects in a physical system.
Q: Can I use the distance formula to find the distance between points in a space with non-standard units?
A: No, the distance formula is used to find the distance between points in a space with standard units. To find the distance between points in a space with non-standard units, you need to use a different formula or method.
Q: What are some common mistakes to avoid when using the distance formula in science and engineering?
A: Some common mistakes to avoid when using the distance formula in science and engineering include:
- Not plugging in the correct values for the coordinates
- Not simplifying the square root correctly
- Not using the correct formula for the distance between two points in a two-dimensional space
Q: Can I use the distance formula to find the distance between points in a space with non-Euclidean geometry?
A: No, the distance formula is used to find the distance between points in a Euclidean space. To find the distance between points in a space with non-Euclidean geometry, you need to use a different formula or method.
Q: What is the relationship between the distance formula and the concept of distance in computer science?
A: The distance formula is used to find the distance between points, which is related to the concept of distance in computer science. The distance formula is used to calculate the distance between two points in a computer graphics or game development context.
Q: Can I use the distance formula to find the distance between points in a space with non-standard coordinates?
A: No, the distance formula is used to find the distance between points in a space with standard coordinates. To find the distance between points in a space with non-standard coordinates, you need to use a different formula or method.
Q: What are some common applications of the distance formula in computer science?
A: Some common applications of the distance formula in computer science include:
- Calculating the distance between two objects in a computer graphics or game development context
- Determining the length of a line segment in a two-dimensional space
- Finding the distance between two points in a machine learning or data analysis context
- Calculating the distance between two objects in a computer vision context