Hey guys! Let's dive into a classic geometry problem where we're given some side lengths and need to figure out another. Specifically, we're going to tackle the question: If AC = 138 and BC = 129, calculate AB. Sounds like fun, right? This type of problem is super common, and understanding how to solve it is a fundamental skill. We'll break down the steps, making sure it's crystal clear. Let's get started. The core idea here is to understand the relationship between the points A, B, and C. They are, at a minimum, collinear, meaning they lie on the same line. Depending on the problem, point B may or may not be between points A and C. In this scenario, we must assume that point B is located between A and C. To find the length of AB, you need to think about the relationship between the segments. Since B is located between A and C, you can use the segment addition postulate. This means that the length of segment AB plus the length of segment BC equals the length of segment AC. The segment addition postulate is a fundamental concept in geometry that helps us understand how segments combine. It's like saying that if you have two smaller pieces of a puzzle, and you put them together, you get a larger piece that is equal to the sum of the two smaller pieces. In this case, our puzzle pieces are the segments, and when we put AB and BC together, we get AC.

Understanding the Problem: Deciphering the Given Information

First things first, let's make sure we totally get what's being asked. We know that AC = 138 and BC = 129. What does this mean? Well, AC represents the total distance from point A to point C, and BC represents the distance from point B to point C. Our goal is to find the distance between points A and B, which we can denote as AB. The first step involves recognizing that the points A, B, and C are collinear, which means they lie on the same straight line. This is a crucial piece of information because it allows us to use the segment addition postulate. The segment addition postulate states that if point B lies between points A and C, then AB + BC = AC. This means that the length of segment AB plus the length of segment BC equals the length of segment AC. Think of it like this: if you walk from A to B and then from B to C, the total distance you walk is the same as walking directly from A to C. This postulate is like a fundamental rule of geometry that allows us to break down complex problems into smaller, more manageable parts. We are trying to find AB, that's what we want to calculate.

Applying the Segment Addition Postulate

Now we can use the segment addition postulate to set up an equation. Since B is between A and C, we know that: AB + BC = AC. We have the values for AC and BC, so we can plug those numbers into the equation: AB + 129 = 138. Now, to solve for AB, we need to isolate it. This means getting AB by itself on one side of the equation. To do that, we'll subtract 129 from both sides of the equation. This gives us: AB = 138 - 129. When we perform the subtraction, we find that AB = 9. So, the distance between points A and B is 9 units. This might seem simple, but understanding the segment addition postulate is key to solving more complex geometry problems. Make sure you understand why we did the subtraction. We wanted to isolate AB, and since 129 was being added to it, we needed to do the opposite operation, which is subtraction. Always double-check your work to make sure your answer makes sense in the context of the problem. If AB were larger than AC, for example, you would know you made a mistake. Always remember to use the correct units. While the problem doesn't specify the units, it's good practice to include them when they are provided.

Calculation and Solution

Alright, let's crunch the numbers. We've got the equation AB = 138 - 129. Doing the subtraction, we get AB = 9. Therefore, the length of segment AB is 9 units. And that's it, we have solved the problem! Congratulations! You've successfully calculated AB using the segment addition postulate. This means that the distance between point A and point B is 9 units. Think of it like this: the total distance from A to C is 138, and a portion of that distance, from B to C, is 129. The remaining distance, from A to B, must be the difference between the two. This type of problem-solving approach is crucial in geometry and other areas of mathematics. By understanding the relationships between segments and applying the right postulates, you can break down complex problems into manageable steps. Now, let's recap the steps. First, we identified the given information: AC = 138 and BC = 129. Next, we understood the segment addition postulate. Then, we set up the equation, plugged in the values, and solved for AB. Finally, we arrived at the solution: AB = 9 units.

Conclusion: Key Takeaways and Next Steps

So, what did we learn? We learned how to solve a geometry problem using the segment addition postulate. We started with the given information, set up an equation, and solved for the unknown. Remember, the key is to understand the relationships between the segments and to apply the correct postulates. Now that you've got this problem under your belt, you're ready for more challenging geometry questions! Keep practicing, and you'll become a pro at these problems in no time. For your next steps, try solving similar problems with different numbers. You can also explore other geometric concepts, such as angles and triangles. Practice makes perfect, guys! Try to apply this logic to other similar problems, like finding the missing side of a triangle or even a more complex shape. Geometry is all about understanding shapes and their properties, so the more you practice, the better you'll get. Don't be afraid to experiment with different problems and challenges. The more you put your skills to the test, the more confident you'll become in your abilities. And hey, if you get stuck, don't worry! That's what learning is all about. Take a break, revisit the concepts, and try again. You've got this!

I hope this explanation was helpful and easy to follow. Geometry can be a lot of fun, and with a little practice, you'll be acing these problems in no time. Keep up the great work, and happy calculating! If you enjoyed this explanation, you can share it with friends and family. This will help them understand this geometrical problem. Do not hesitate to ask for help from professors if you encounter more difficult problems.