Understanding the Distance Between Coordinates in Geometry

When dealing with coordinates in a two-dimensional plane, it's essential to understand how to measure the distance between two points. This article will explore the process of calculating the distance between the coordinates (-7, 2) and (5, -3) using the Pythagorean Theorem. We will break down the steps involved in this calculation and explain why the result is meaningful.

Introduction to Coordinate Geometry

Coordinate geometry is a branch of mathematics that uses the coordinate plane to represent and solve geometric problems. Each point on the coordinate plane is defined by an ordered pair (x, y), where x represents the horizontal position (left or right) and y represents the vertical position (up or down).

Calculating the Distance Between Two Coordinates

Consider the two coordinates (-7, 2) and (5, -3). To find the distance between these points, we can use the distance formula, which is derived from the Pythagorean Theorem. The distance formula is as follows:

d √[(x? - x?)2 (y? - y?)2]

Here, (x?, y?) and (x?, y?) are the coordinates of the two points.

Step 1: Calculate the X-Distance

The X-distance is the difference in the x-coordinates of the two points. In this case, we need to find the difference between x? -7 and x? 5.

X-distance 5 - (-7) 5 7 12

Step 2: Calculate the Y-Distance

The Y-distance is the difference in the y-coordinates of the two points. Here, we need to find the difference between y? 2 and y? -3.

Y-distance 2 - (-3) 2 3 5

Step 3: Apply the Pythagorean Theorem

Now, we can use the Pythagorean Theorem to find the actual distance between the two points. The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's denote the X-distance as a and the Y-distance as b. The distance (d) is calculated as:

d √(a2 b2)

Substituting the values we found:

d √(122 52) √(144 25) √169 13

Pythagorean Triple

Notably, the values 5, 12, and 13 form a Pythagorean triple. A Pythagorean triple consists of three positive integers a, b, and c, such that a2 b2 c2. In this case, 52 122 169 132.

Significance of the Distance Calculation

The distance calculation between two coordinates is not just a mathematical exercise; it has practical applications in various fields. For example, in computer graphics, it's essential to calculate distances to draw shapes accurately. In GPS technology, the distances between coordinates are crucial for navigation and mapping.

Practical Examples and Applications

Let's apply this knowledge to a real-world example. Suppose you are designing a game and need to know the distance between two points on a grid. By using the distance formula, you can determine the path length a character needs to travel to move from one point to another.

In the context of architecture, knowing the distance between two points can help in laying out the building structure or estimating the distance between different parts of a building.

Conclusion

In summary, the distance between the coordinates (-7, 2) and (5, -3) is 13 units. This calculation is done using the distance formula, which involves the differences in the x and y coordinates and the application of the Pythagorean Theorem. Understanding these concepts helps in various applications, from computer graphics to practical geometry problems.