Formula For Distance

Interactive and Dynamic Lesson to Find the Distance Between two Points

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\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Distance Between Two Points

Understanding the distance between two points is a foundational concept in coordinate geometry that not only underpins various geometric calculations but also has practical applications in fields like engineering, physics, computer graphics, and navigation. Let's delve deeper into what this distance represents, how it's calculated, and why it's important.

At its core, the distance between two points refers to the shortest or straight-line path connecting them in a plane. Imagine placing two distinct points on a flat surface, like a piece of paper or a computer screen. The distance between these points is akin to stretching a tight string between them—the length of that string represents the distance.

The Distance Formula

Given two points in a Cartesian plane, \( P_1 (x_1, y_1) \) and \( P_2 (x_2, y_2) \), the distance \( D \) between them can be calculated using the following formula:

\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

X Y (x₁,y₁) (x₂,y₂) Distance

Explanation of the Formula:

  1. Difference in Coordinates:
    • \( (x_2 - x_1) \): The horizontal distance between the two points.
    • \( (y_2 - y_1) \): The vertical distance between the two points.

  2. Squaring the Differences:

    Squaring ensures that the distance is always a positive value, regardless of the direction between the points.

  3. Summing the Squares:

    Adding the squares of the differences accounts for both the horizontal and vertical distances.

  4. Square Root:

    Taking the square root of the sum gives the actual straight-line distance between the two points.

This formula is derived from the Pythagorean Theorem, which relates the sides of a right triangle.

x2 - x1 and y2 - y1 are the lengths of the legs (the sides that make up a right angle).

d is the length of the hypotenuse.

Applying the Pythagorean theore, we get d2 = (x2 - x1)2 + (y2 - y1)2

Then, you just need to take the square root of both sides to get the formula for distance.

Step-by-Step Guide to Using the Distance Formula

  1. Identify the Coordinates:

    Determine the coordinates of the two points between which you want to find the distance.

  2. Subtract the Coordinates:

    Subtract the \( x \)-coordinates and the \( y \)-coordinates of the two points.

  3. Square the Differences:

    Square both results from the subtraction.

  4. Add the Squared Differences:

    Add the two squared numbers together.

  5. Take the Square Root:

    Calculate the square root of the sum to find the distance.

Examples

Let's apply the distance formula through a series of examples to solidify your understanding.

Example 1: Finding the Distance Between (2, 3) and (6, 6)

Given Points:

  • \( P_1 (x_1, y_1) = (2, 3) \)
  • \( P_2 (x_2, y_2) = (6, 6) \)
X Y (2,3) (6,6) d = 5

Applying the Distance Formula:

\[ \begin{align*} D &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(6 - 2)^2 + (6 - 3)^2} \\ &= \sqrt{(4)^2 + (3)^2} \\ &= \sqrt{16 + 9} \\ &= \sqrt{25} \\ &= 5 \end{align*} \]

Result: The distance between the points (2, 3) and (6, 6) is 5 units.

Example 2: Finding the Distance Between (17, 12) and (9, 6)

Given Points:

  • \( P_1 (x_1, y_1) = (17, 12) \)
  • \( P_2 (x_2, y_2) = (9, 6) \)

Applying the Distance Formula:

\[ \begin{align*} D &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(9 - 17)^2 + (6 - 12)^2} \\ &= \sqrt{(-8)^2 + (-6)^2} \\ &= \sqrt{64 + 36} \\ &= \sqrt{100} \\ &= 10 \end{align*} \]

Result: The distance between the points (17, 12) and (9, 6) is 10 units.

Example 3: Finding the Distance Between (7, 8) and (4, 5)

Given Points:

  • \( P_1 (x_1, y_1) = (7, 8) \)
  • \( P_2 (x_2, y_2) = (4, 5) \)

Applying the Distance Formula:

\[ \begin{align*} D &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(4 - 7)^2 + (5 - 8)^2} \\ &= \sqrt{(-3)^2 + (-3)^2} \\ &= \sqrt{9 + 9} \\ &= \sqrt{18} \\ &= \sqrt{9 \times 2} \\ &= 3\sqrt{2} \\ &\approx 4.24 \end{align*} \]

Result: The distance between the points (7, 8) and (4, 5) is \( 3\sqrt{2} \) units, which is approximately 4.24 units.

Practical Applications

Understanding the distance formula is not only essential for academic purposes but also has practical applications in various fields:

  • Navigation: Calculating the shortest path between two locations.
  • Engineering: Determining distances between structural points.
  • Computer Graphics: Rendering distances between pixels or objects.
  • Geography: Measuring distances on maps.

Tips for Success

  • Double-Check Coordinates: Ensure you correctly identify and substitute the \( x \) and \( y \) coordinates of both points.
  • Handle Negative Differences Carefully: Remember that squaring a negative number results in a positive value.
  • Simplify Square Roots: If possible, simplify the square root for an exact answer. Otherwise, use a calculator for decimal approximations.
  • Practice: Work through multiple examples to become comfortable with the formula and its applications.

Centroid of a triangle