Wondering about the distance between two points in a Cartesian plane? Let’s dive into the calculation for finding the distance between the points (3, 12) and (14, 2).

## What Is the Distance Between the Points (3, 12) And (14, 2)?

To determine the distance between two points in a coordinate system, we can apply the Pythagorean theorem. This theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of its two legs.

Applying this concept to our scenario, we can consider each coordinate as a leg of a right triangle. By subtracting the x-coordinates and y-coordinates respectively, we get 11 as one leg and -10 as another leg. Taking their squares, we have 121 and 100.

Adding these squares together and taking their square root gives us √(121 + 100) = √221 ≈ 14.87 units. Therefore, the distance between (3, 12) and (14, 2) is approximately 14.87 units.

Remember that calculating distances using this method works for any pair of coordinates on a plane. So now you’re equipped with knowledge to find distances efficiently!

## Understanding the Distance Formula

When it comes to finding the distance between two points on a coordinate plane, we can rely on a handy tool called the Distance Formula. This formula allows us to determine the straight-line distance between any two given points, regardless of their positions in the Cartesian coordinate system.

The Distance Formula is based on the Pythagorean Theorem, which states that in a right 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. By applying this theorem to our coordinates, we can calculate distances accurately.

## Applying the Distance Formula to Cartesian Coordinates

To apply the Distance Formula, let’s take a closer look at how it works with Cartesian coordinates. In our case, we want to find out what is the distance between two points: (3, 12) and (14, 2). Let’s denote these points as A(x1, y1) = (3, 12) and B(x2, y2) = (14, 2).

The formula itself states that d(A,B) = √((x2 – x1)^2 + (y2 – y1)^2), where d(A,B) represents our desired distance.

So for our example:

d(A,B) = √((14 – 3)^2 + (2 – 12)^2) = √(11^2 + (-10)^2) = √(121 + 100) = √221 ≈ 14.87 units

Hence, using this formula and calculations we’ve made above, we find that distance between point A(3 ,12) and point B(14 , 2) is approximately 14.87 units.

## Breaking Down The Points in The Given Problem

Now, let’s break down the points in our given problem to better understand how they contribute to the distance calculation.

Point A(3, 12): The x-coordinate of this point is 3, and the y-coordinate is 12. This means that if we were to plot this point on a graph, it would lie 3 units to the right on the x-axis and 12 units up on the y-axis.

Point B(14, 2): Similarly, for point B, we have an x-coordinate of 14 and a y-coordinate of 2. This tells us that when plotted on a graph, point B would be situated 14 units to the right on the x-axis and only 2 units up on the y-axis.

By understanding and visualizing these coordinates, we can see how they come together in our distance formula calculation. The difference between their respective x-coordinates contributes to the horizontal component of our distance calculation, while the difference between their y-coordinates determines its vertical component.

## Identifying the X and Y Coordinates

To find the distance between two points, (3, 12) and (14, 2), we first need to identify the X and Y coordinates of each point. The X coordinate represents the horizontal position on a graph, while the Y coordinate represents the vertical position.

For the point (3, 12), the X coordinate is 3 and the Y coordinate is 12. This means that this point is located at 3 units to the right of the origin (0 on the X-axis) and 12 units above it.

Similarly, for the point (14, 2), the X coordinate is 14 and the Y coordinate is 2. This implies that this point is situated at 14 units to the right of the origin and 2 units above it.

Now that we have identified both sets of coordinates, we can proceed to calculate the distance between these two points using a formula called Euclidean distance or Pythagorean theorem. The formula states that:

Distance = √((X₂ – X₁)² + (Y₂ – Y₁)²)

Plugging in our values from earlier:

Distance = √((14 – 3)² + (2 – 12)²) = √(11² + (-10)²) = √(121 +100) = √221 ≈ 14.87

Therefore, based on our calculations, we can conclude that the distance between points (3, 12) and (14, 2) is approximately equal to 14.87 units.

Please note that these calculations assume a Cartesian plane with no additional factors such as slopes or angles involved. Keep in mind that accurately identifying coordinates is crucial when calculating distances between points in various mathematical applications or real-world scenarios.