Distance Between (12.9, 14) And (2, -6.5): A Step-by-Step Guide

Hey math enthusiasts! Ever found yourself wondering how to calculate the distance between two points on a coordinate plane? It's a fundamental concept in geometry, and mastering it opens doors to solving various problems in math and real-life scenarios. In this guide, we'll walk you through the process step-by-step, using the points (12.9, 14) and (2, -6.5) as our example. So, grab your calculators and let's dive in!

Understanding the Distance Formula

The distance formula is the cornerstone of calculating the distance between two points in a coordinate plane. It’s derived from the Pythagorean theorem, a² + b² = c², which relates the sides of a right triangle. Think of the two points as vertices of a right triangle, where the distance between them is the hypotenuse. The formula is expressed as:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

• (x₁, y₁) are the coordinates of the first point.
• (x₂, y₂) are the coordinates of the second point.

In simpler terms, we find the difference in the x-coordinates, square it, find the difference in the y-coordinates, square that, add the two squared values, and then take the square root of the sum. Sounds like a mouthful, but it’s quite straightforward once you break it down. Let's understand the formula in detail:

First, identify the coordinates of your two points. In our case, we have point 1 as (12.9, 14) and point 2 as (2, -6.5). This step is crucial because the entire calculation hinges on these values. A small error here can throw off the whole result. Think of these points as locations on a map; each coordinate tells you how far to move along the horizontal (x) and vertical (y) axes from the origin (0,0). By accurately pinpointing these locations, we set the stage for a precise distance calculation.

Next, calculate the difference in x-coordinates (x₂ - x₁) and y-coordinates (y₂ - y₁). For the x-coordinates, we subtract the x-value of point 1 from the x-value of point 2, which gives us 2 - 12.9 = -10.9. Similarly, for the y-coordinates, we subtract the y-value of point 1 from the y-value of point 2, resulting in -6.5 - 14 = -20.5. The order of subtraction is important, but because we square these differences in the next step, the sign (+ or -) won't ultimately affect our final answer. However, maintaining consistency in subtraction (always subtracting the first point from the second, or vice versa) helps avoid confusion.

Then, square each of these differences. Squaring a number means multiplying it by itself. So, (-10.9)² equals -10.9 multiplied by -10.9, which is 118.81. Likewise, (-20.5)² equals -20.5 multiplied by -20.5, resulting in 420.25. Squaring eliminates any negative signs, ensuring we're dealing with positive distances. This step is rooted in the Pythagorean theorem, where we are essentially finding the squares of the lengths of the two legs (sides) of our right triangle.

Now, add the squared differences together. We add the results from the previous step: 118.81 + 420.25, which gives us 539.06. This sum represents the square of the distance between our two points. In the context of the Pythagorean theorem, this sum is equivalent to c², where c is the length of the hypotenuse (the distance we're trying to find).

Finally, take the square root of the sum. This is the last step in unraveling the distance. We take the square root of 539.06, which yields approximately 23.217666. Taking the square root reverses the squaring operation, allowing us to find the actual distance. This step essentially completes the application of the Pythagorean theorem, revealing the length of the hypotenuse of our right triangle, which is the distance between the two points.

Applying the Formula to Our Points (12.9, 14) and (2, -6.5)

Let's apply the distance formula to our specific points, (12.9, 14) and (2, -6.5). We'll break it down step by step to ensure clarity.

1. Identify the coordinates:

   • (x₁, y₁) = (12.9, 14)
   • (x₂, y₂) = (2, -6.5)

2. Calculate the difference in x-coordinates:

   • x₂ - x₁ = 2 - 12.9 = -10.9

3. Calculate the difference in y-coordinates:

   • y₂ - y₁ = -6.5 - 14 = -20.5

4. Square the differences:

   • (-10.9)² = 118.81
   • (-20.5)² = 420.25

5. Add the squared differences:

   • 118.81 + 420.25 = 539.06

6. Take the square root of the sum:

   • √539.06 ≈ 23.217666

Rounding to the Nearest Tenth

The question asks us to round our answer to the nearest tenth. Looking at our result, 23.217666, the digit in the tenths place is 2. The digit to its right (in the hundredths place) is 1, which is less than 5. Therefore, we round down and keep the 2 in the tenths place. So, the distance between the points (12.9, 14) and (2, -6.5), rounded to the nearest tenth, is 23.2.

Rounding is a critical skill in mathematics and everyday life. It allows us to simplify numbers, making them easier to work with and understand. When rounding to the nearest tenth, we're essentially asking,