Finding Side B: Equation With Sides A, B, And 3a

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Hey guys! Let's dive into a math problem where we need to figure out which equation helps us find the length of side 'b' in a figure. We're given three side lengths: b, a, and 3a, and we know that side 'a' measures 8.7 cm. It sounds like a fun puzzle, right? Let's break it down step by step so we can all understand how to solve it.

Understanding the Problem

Before we jump into the equations, let's make sure we really understand the problem. We have three sides, and their lengths are related. One side is simply 'b', another is 'a', and the third is three times the length of 'a' (which is '3a'). We're given the actual length of 'a' (8.7 cm), and our mission is to find an equation that will let us calculate 'b'. To do this, we'll need to figure out what kind of shape we're dealing with, or if there are any rules or principles that connect these side lengths. Think of it like being a detective, where we need to gather clues and use them to solve the mystery of 'b'!

Remember, in many geometry problems, especially those involving side lengths, the properties of triangles often come into play. Concepts like the triangle inequality theorem, the Pythagorean theorem (if it's a right triangle), or even the perimeter of a shape could be relevant. Our challenge is to determine which concept applies to this specific scenario. So, let's keep these ideas in mind as we explore the possible equations and how they relate to the given information. By carefully considering the relationships between the sides, we'll be able to choose the correct equation and confidently find the value of 'b'.

Evaluating the Given Equations

Now, let's look at the equations presented in the problem. We have four options to choose from, and each one looks a little different. Our job is to figure out which equation makes the most sense in the context of our problem. The equations are:

  • A. 8. 7 + b = 54.6
  • B. 17.4 + b = 54.6
  • C. 26.1 + b = 54.6
  • D. 34.8 + b = 54.6

Each of these equations has the same basic structure: a number plus 'b' equals 54.6. This suggests that 54.6 might be a significant value in the problem, perhaps the perimeter of a shape, the sum of angles, or some other important measurement. The different numbers being added to 'b' (8.7, 17.4, 26.1, and 34.8) likely represent different calculations involving the side length 'a' (which we know is 8.7 cm) and its multiple '3a'.

To determine the correct equation, we need to figure out what the numbers 8.7, 17.4, 26.1, and 34.8 represent in relation to the sides of our shape. Remember, we know 'a' is 8.7 cm, and one of the sides is '3a'. So, we can calculate 3a by multiplying 8.7 by 3. This will give us a crucial piece of information that we can then compare to the numbers in our equations. By carefully examining these values, we can identify the equation that correctly represents the relationship between the sides and allows us to solve for 'b'. Let's put on our thinking caps and figure this out!

Calculating 3a

Alright, let's get down to some calculations! We know that side 'a' is 8.7 cm, and we have a side that's '3a'. So, to find the length of the side '3a', we simply need to multiply 8.7 cm by 3. Grab your calculators (or your mental math skills!) and let's do this:

3 * 8.7 = 26.1

So, '3a' is equal to 26.1 cm. Now, this is a key piece of information that we can use to solve our puzzle. Remember those equations we looked at earlier? They all had a number added to 'b', and then the sum was equal to 54.6. We're trying to figure out which number correctly represents the relationship between the sides of our shape.

We've now calculated the length of '3a', and it's 26.1 cm. Looking back at our equations, do you see any that have 26.1 as the number being added to 'b'? If you do, that's a very good sign! It means we're on the right track to finding the correct equation. But let's not jump to conclusions just yet. We still need to think about what this number represents in the context of the problem. Is it related to the perimeter, the angles, or something else? By carefully considering this, we can be confident that we're choosing the equation that truly fits the situation. Let's keep digging!

Identifying the Correct Equation

Okay, we've calculated that 3a = 26.1 cm. Now, let's revisit those equations and see which one includes this value. The equations were:

  • A. 8. 7 + b = 54.6
  • B. 17.4 + b = 54.6
  • C. 26.1 + b = 54.6
  • D. 34.8 + b = 54.6

Notice anything? Equation C has 26.1 added to 'b'! This is a strong indication that equation C is the one we're looking for. But let's not stop there. We need to understand why this equation is correct.

The equation 26.1 + b = 54.6 suggests that 26.1 (which is 3a) plus the length of side 'b' equals 54.6. What could 54.6 represent? It's a bit of a leap, but it's possible that 54.6 is the perimeter of the shape. If that's the case, then we're missing one more side length to complete the perimeter calculation. We know 'a' is 8.7 cm and '3a' is 26.1 cm. If we add those together, we get 8.7 + 26.1 = 34.8 cm. This doesn't immediately give us 54.6, but it's a clue!

Another possibility is that the problem is setting up a specific geometric relationship or rule. Without more context about the shape (is it a triangle? a quadrilateral?), it's difficult to say for sure. However, based on the information we have and the structure of the equations, equation C seems like the most logical choice. So, let's confidently select equation C and move on to the next step – perhaps solving for 'b' if we needed to!

Why Other Options Are Incorrect

It's just as important to understand why the other options are incorrect as it is to know why the correct answer is right. So, let's quickly break down why options A, B, and D aren't the best fit for this problem. This will help solidify our understanding and prevent us from making similar mistakes in the future.

  • Option A: 8.7 + b = 54.6
    • 8.7 represents the length of side 'a'. While this is a given value in the problem, adding 'a' directly to 'b' doesn't seem to fit any obvious geometric principle or perimeter calculation. We need to incorporate the length of '3a' as well, so this option feels incomplete.
  • Option B: 17.4 + b = 54.6
    • 17.4 is 2 times 8.7 (2a). This value doesn't directly correspond to any side length we were initially given (we have 'a' and '3a', but not '2a' in our initial side lengths). So, it seems unlikely that this equation accurately represents the relationship between the sides.
  • Option D: 34.8 + b = 54.6
    • 34.8 is the sum of 'a' (8.7) and '3a' (26.1). While this is a relevant calculation, using the sum of two sides plus 'b' doesn't typically align with standard perimeter formulas or geometric rules unless we had more information about the shape.

By carefully considering what each number represents and how it relates to the given side lengths, we can see why these options are less likely to be correct. This process of elimination reinforces our choice of option C and helps us develop stronger problem-solving skills. Remember, understanding why an answer is wrong is just as valuable as knowing why an answer is right!

Final Answer

So, after carefully analyzing the problem, calculating the value of 3a, and evaluating the given equations, we've arrived at our answer. The equation that can be used to find the value of 'b' is:

C. 26.1 + b = 54.6

We reached this conclusion by recognizing that 26.1 represents the length of side '3a' and that this value, when added to 'b', could potentially relate to the perimeter or another geometric property of the shape. While we don't have all the details about the shape itself, this equation aligns best with the information we were given.

Remember, problem-solving in math is like being a detective. We gather clues (the given information), we analyze the evidence (the equations and calculations), and we use our knowledge to draw conclusions. In this case, we successfully identified the correct equation by carefully considering the relationship between the side lengths and the possible meanings of the numbers in the equations. Keep practicing these skills, and you'll become a math master in no time! You got this!