Calculating Distance Between Points A Comprehensive Guide

by Scholario Team 58 views

Hey guys! Ever wondered how to find the exact distance between two points on a graph? It's a fundamental concept in mathematics, and we're going to break it down today. We'll tackle some specific examples and equip you with the tools to solve these problems like a pro.

Understanding the Distance Formula

Before we dive into the exercises, let's quickly recap the distance formula. This formula is your best friend when it comes to calculating the distance between two points in a coordinate plane. It's derived from the Pythagorean theorem (remember that from geometry?), and it looks like this:

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 simple terms, the distance formula calculates the length of the hypotenuse of a right triangle formed by the two points. The difference in x-coordinates gives you the length of one leg, the difference in y-coordinates gives you the length of the other leg, and the distance formula puts it all together to find the hypotenuse (which is the distance between the points!).

Now that we've refreshed our memory on the distance formula, let's jump into the exercises and see how it works in practice. We'll go through each problem step-by-step, making sure you grasp the concept thoroughly. Remember, practice makes perfect, so the more you work with the distance formula, the easier it will become.

Exercise 11: Finding the Distance Between A(6, 8) and B(-1, 8)

Okay, let's kick things off with our first problem. We need to find the distance between points A(6, 8) and B(-1, 8). The key here is to carefully identify our x and y coordinates for each point. Let's label them:

  • A(6, 8): x₁ = 6, y₁ = 8
  • B(-1, 8): x₂ = -1, y₂ = 8

Now that we have our coordinates, we can plug them directly into the distance formula:

Distance = √[(-1 - 6)² + (8 - 8)²]

Let's break this down step by step. First, we calculate the differences inside the parentheses:

Distance = √[(-7)² + (0)²]

Next, we square the results:

Distance = √[49 + 0]

Then, we add the squared values:

Distance = √49

Finally, we take the square root:

Distance = 7

So, the distance between points A and B is 7 units. Notice something interesting here: the y-coordinates of both points are the same (8). This means that the line segment AB is horizontal. In such cases, you could also simply find the distance by taking the absolute difference of the x-coordinates: |6 - (-1)| = 7. However, the distance formula works perfectly fine too!

Exercise 12: Calculating the Distance Between C(5, -6) and D(5, 6)

Alright, let's move on to the next one! This time, we want to find the distance between points C(5, -6) and D(5, 6). Just like before, the first step is to identify our coordinates:

  • C(5, -6): x₁ = 5, y₁ = -6
  • D(5, 6): x₂ = 5, y₂ = 6

Now, let's plug these values into the distance formula:

Distance = √[(5 - 5)² + (6 - (-6))²]

Let's simplify step by step:

Distance = √[(0)² + (12)²]

Distance = √[0 + 144]

Distance = √144

Distance = 12

Therefore, the distance between points C and D is 12 units. Similar to the previous example, we can observe that the x-coordinates of both points are the same (5). This indicates that the line segment CD is vertical. You could also find the distance by taking the absolute difference of the y-coordinates: |6 - (-6)| = 12. Again, the distance formula provides a reliable method regardless!

Exercise 13: Determining the Distance Between E(-2, 0) and F(11, 0)

Let's keep the momentum going! Our next challenge is to find the distance between points E(-2, 0) and F(11, 0). You know the drill by now – let's start by identifying those coordinates:

  • E(-2, 0): x₁ = -2, y₁ = 0
  • F(11, 0): x₂ = 11, y₂ = 0

Time to plug those values into our trusty distance formula:

Distance = √[(11 - (-2))² + (0 - 0)²]

Let's simplify this step by step:

Distance = √[(13)² + (0)²]

Distance = √[169 + 0]

Distance = √169

Distance = 13

So, the distance between points E and F is 13 units. Notice that the y-coordinates are both 0 in this case. This means that both points lie on the x-axis. The distance is simply the absolute difference of their x-coordinates: |11 - (-2)| = 13. The distance formula, as always, provides the correct answer.

Exercise 14: Finding the Distance Between Q(1, -5) and T(9, 1)

Okay, we've reached our final problem! Let's find the distance between points Q(1, -5) and T(9, 1). Let’s follow the now-familiar pattern and start by identifying our coordinates:

  • Q(1, -5): x₁ = 1, y₁ = -5
  • T(9, 1): x₂ = 9, y₂ = 1

Let's plug these values into the distance formula and see what we get:

Distance = √[(9 - 1)² + (1 - (-5))²]

Now, let's simplify step by step:

Distance = √[(8)² + (6)²]

Distance = √[64 + 36]

Distance = √100

Distance = 10

Therefore, the distance between points Q and T is 10 units. In this case, neither the x-coordinates nor the y-coordinates are the same, so the line segment QT is neither horizontal nor vertical. The distance formula is essential here to find the correct distance.

Conclusion: Mastering the Distance Formula

Great job, guys! We've successfully calculated the distance between several pairs of points using the distance formula. Remember, the key is to carefully identify the coordinates of each point and then plug them into the formula. By breaking down the problem into smaller steps, you can easily solve these types of exercises.

The distance formula is a fundamental concept in coordinate geometry, and mastering it will help you in many other areas of mathematics. Keep practicing, and you'll become a distance-calculating whiz in no time!

If you have any further questions or want to explore more challenging problems, feel free to ask! Happy calculating!