Find The Distance Between Each Pair Of Points Answer Key

Mastering the Distance Formula: A Comprehensive Guide to Finding the Distance Between Two Points

Finding the distance between two points is a fundamental concept in geometry and has wide applications in various fields, from computer graphics to navigation. This guide provides a comprehensive exploration of the distance formula, breaking down its mathematical foundation, offering step-by-step instructions for its application, and providing illustrative examples. Whether you are a student grappling with geometry or a professional seeking a refresher, this article will equip you with the tools and understanding to confidently calculate distances between points.

Introduction: The Importance of Distance Calculation

The concept of distance is intuitive, but its precise calculation in a coordinate system relies on the distance formula. This formula, derived from the Pythagorean theorem, allows us to determine the length of a line segment connecting two points in a plane or in three-dimensional space. Understanding how to use the distance formula is essential for:

• Geometry: Solving problems involving triangles, quadrilaterals, and other geometric shapes.
• Coordinate Geometry: Analyzing the properties of geometric figures defined by coordinates.
• Calculus: Calculating arc lengths and distances along curves.
• Physics: Determining the displacement of objects and calculating distances in motion problems.
• Computer Graphics: Calculating distances between objects in a virtual environment.
• Navigation: Determining distances between locations using GPS coordinates.

The Distance Formula: Unveiling the Mathematical Foundation

The distance formula is rooted in the Pythagorean theorem, which states that in a right 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 (legs). Mathematically, this is expressed as:

a² + b² = c²

Where:

• a and b are the lengths of the legs of the right triangle.
• c is the length of the hypotenuse.

Now, consider two points in a two-dimensional coordinate plane, (x₁, y₁) and (x₂, y₂). We can form a right triangle by drawing horizontal and vertical lines from these points. The horizontal line will have a length equal to the difference in the x-coordinates, |x₂ - x₁|, and the vertical line will have a length equal to the difference in the y-coordinates, |y₂ - y₁|. The distance between the two points is then the length of the hypotenuse of this right triangle.

Applying the Pythagorean theorem, we get:

distance² = (x₂ - x₁)² + (y₂ - y₁)²

Taking the square root of both sides, we arrive at the distance formula:

distance = √((x₂ - x₁)² + (y₂ - y₁)²)

This formula can be extended to three-dimensional space by adding a term for the difference in the z-coordinates:

distance = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

Step-by-Step Guide: Applying the Distance Formula

Using the distance formula is straightforward, but it requires careful attention to detail. Here's a step-by-step guide to help you apply it correctly:

1. Identify the Coordinates: Determine the coordinates of the two points between which you want to find the distance. Let's call them (x₁, y₁) and (x₂, y₂). For example, point A might be (2, 3) and point B might be (5, 7).

2. Label the Coordinates: Clearly label the x and y coordinates for each point. This helps avoid confusion when substituting values into the formula. So, x₁ = 2, y₁ = 3, x₂ = 5, and y₂ = 7.

3. Substitute into the Formula: Plug the coordinates into the distance formula:

   distance = √((x₂ - x₁)² + (y₂ - y₁)²)

   Using our example:

   distance = √((5 - 2)² + (7 - 3)²)

4. Simplify the Expression: Simplify the expression inside the parentheses first:

   distance = √((3)² + (4)²)

5. Square the Terms: Square each of the terms inside the square root:

   distance = √(9 + 16)

6. Add the Squared Terms: Add the squared terms together:

   distance = √(25)

7. Calculate the Square Root: Calculate the square root of the sum to find the distance:

   distance = 5

Therefore, the distance between points A(2, 3) and B(5, 7) is 5 units.

Extending to Three Dimensions:

If you are working in three-dimensional space, the process is similar. Follow the same steps, but include the z-coordinates in the formula:

distance = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

For example, to find the distance between points (1, 2, 3) and (4, 5, 6):

distance = √((4 - 1)² + (5 - 2)² + (6 - 3)²) distance = √((3)² + (3)² + (3)²) distance = √(9 + 9 + 9) distance = √(27) distance = 3√3

Illustrative Examples: Putting the Distance Formula into Practice

Let's work through several examples to solidify your understanding of the distance formula.

Example 1: Finding the Distance Between Two Points in a Plane

Find the distance between the points (-1, 4) and (3, 1).

1. Identify Coordinates: (x₁, y₁) = (-1, 4), (x₂, y₂) = (3, 1)
2. Label Coordinates: x₁ = -1, y₁ = 4, x₂ = 3, y₂ = 1
3. Substitute into Formula: distance = √((3 - (-1))² + (1 - 4)²)
4. Simplify: distance = √((4)² + (-3)²)
5. Square: distance = √(16 + 9)
6. Add: distance = √(25)
7. Square Root: distance = 5

Therefore, the distance between the points (-1, 4) and (3, 1) is 5 units.

Example 2: Finding the Distance Between Two Points in Three-Dimensional Space

Find the distance between the points (2, -3, 1) and (0, 4, -2).

1. Identify Coordinates: (x₁, y₁, z₁) = (2, -3, 1), (x₂, y₂, z₂) = (0, 4, -2)
2. Label Coordinates: x₁ = 2, y₁ = -3, z₁ = 1, x₂ = 0, y₂ = 4, z₂ = -2
3. Substitute into Formula: distance = √((0 - 2)² + (4 - (-3))² + (-2 - 1)²)
4. Simplify: distance = √((-2)² + (7)² + (-3)²)
5. Square: distance = √(4 + 49 + 9)
6. Add: distance = √(62)
7. Square Root: distance = √62 ≈ 7.87

Therefore, the distance between the points (2, -3, 1) and (0, 4, -2) is approximately 7.87 units.

Example 3: Applying the Distance Formula to Determine Triangle Properties

The vertices of a triangle are A(1, 2), B(4, 6), and C(7, 2). Determine the type of triangle (e.g., equilateral, isosceles, scalene).

To determine the type of triangle, we need to find the lengths of all three sides using the distance formula:

• Distance AB: √((4 - 1)² + (6 - 2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5
• Distance BC: √((7 - 4)² + (2 - 6)²) = √(3² + (-4)²) = √(9 + 16) = √25 = 5
• Distance AC: √((7 - 1)² + (2 - 2)²) = √(6² + 0²) = √36 = 6

Since AB = BC, the triangle is an isosceles triangle.

Common Mistakes and How to Avoid Them

While the distance formula is relatively simple, there are several common mistakes that students often make. Here's how to avoid them:

• Incorrectly Identifying Coordinates: Make sure you correctly identify and label the x and y (or x, y, and z) coordinates of each point. Double-check your work to avoid errors.

• Incorrect Substitution: When substituting the coordinates into the formula, be careful to place them in the correct positions. Pay close attention to the order of subtraction.

• Forgetting to Square: Remember to square the differences between the x-coordinates and y-coordinates (and z-coordinates in three dimensions) before adding them.

• Incorrectly Handling Negative Signs: Be careful when dealing with negative signs. Remember that subtracting a negative number is the same as adding a positive number. Use parentheses to avoid confusion.

• Forgetting the Square Root: Don't forget to take the square root of the sum of the squared terms to find the final distance.

• Not Simplifying Completely: Simplify the expression under the square root as much as possible before calculating the square root. This can make the calculation easier and reduce the chance of errors.

• Units of Measurement: Always remember to include the appropriate units of measurement in your answer if they are provided in the problem (e.g., meters, feet, inches). If no units are specified, you can simply say "units."

Advanced Applications and Extensions

The distance formula is a powerful tool that can be used in a variety of advanced applications. Here are a few examples:

• Finding the Equation of a Circle: The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r². This equation is based on the distance formula, as it states that the distance between any point (x, y) on the circle and the center (h, k) is equal to the radius r.

• Finding the Equation of a Sphere: Similarly, the equation of a sphere with center (h, k, l) and radius r is (x - h)² + (y - k)² + (z - l)² = r².

• Calculating Arc Length: In calculus, the distance formula is used to approximate the length of a curve by dividing it into small line segments and summing the lengths of these segments. This leads to the concept of arc length, which is the integral of the square root of 1 plus the square of the derivative of the function defining the curve.

• Clustering Algorithms: In data science, the distance formula is used in clustering algorithms, such as k-means clustering, to group data points that are close to each other.

• Nearest Neighbor Search: The distance formula is also used in nearest neighbor search algorithms to find the data point that is closest to a given query point.

FAQs About the Distance Formula

Here are some frequently asked questions about the distance formula:

Q: What is the difference between distance and displacement?

A: Distance is the total length of the path traveled by an object, while displacement is the shortest distance between the initial and final positions of the object. Displacement is a vector quantity, meaning it has both magnitude and direction, while distance is a scalar quantity, meaning it has only magnitude.

Q: Can the distance formula be used in non-Euclidean spaces?

A: The distance formula as we have discussed it applies to Euclidean space. In non-Euclidean spaces, such as spherical geometry or hyperbolic geometry, different distance formulas are used.

Q: Does it matter which point I choose as (x₁, y₁) and which I choose as (x₂, y₂)?

A: No, it does not matter. Because you are squaring the difference between the coordinates, the order of subtraction does not affect the result. (a - b)² = (b - a)².

Q: What if the distance is zero?

A: If the distance between two points is zero, it means that the two points are the same point.

Q: Can I use the distance formula to find the distance between a point and a line?

A: No, the distance formula calculates the distance between two points. To find the distance between a point and a line, you need to use a different formula that involves the equation of the line and the coordinates of the point.

Conclusion: The Power of Precision in Distance Measurement

The distance formula is a cornerstone of geometry and a vital tool for numerous applications in mathematics, science, and engineering. By understanding its derivation, mastering its application, and avoiding common pitfalls, you can confidently calculate distances between points in both two and three dimensions. This knowledge not only enhances your problem-solving skills in academic settings but also equips you to tackle real-world challenges that require precise distance measurements. Practice using the formula with a variety of examples to solidify your understanding and unlock its full potential.