Finding DE: A Geometry Problem Explained

Oct 17, 2025 by Dimemap Team 41 views

Hey guys! Let's dive into a fun geometry problem. We're given a diagram, and our mission is to find the length of a line segment, $DE$ . Don't worry, it's not as scary as it sounds! We'll break it down step-by-step and make sure we understand everything. This is a classic example of how geometry problems can be solved using proportions and similar triangles. So, get ready to flex those math muscles! We'll be using the information provided – $LA = 10\text{ cm}$ and $AC = 12\text{ cm}$ – to crack the code and find the length of $DE$ . By the end of this, you'll be a geometry whiz, ready to tackle similar problems with confidence. It's all about understanding the relationships between the different parts of the shapes, and using that knowledge to solve for the unknown. Ready to jump in? Let's go! Remember, geometry is all about shapes, angles, and their relationships. We'll be using this knowledge to solve for the length of $DE$ .

This kind of problem is super common in math. They are designed to test your knowledge of similar triangles and how their sides relate to each other. We will go through the proper steps to solve this problem.

Understanding the Problem: The Basics

Alright, let's get our bearings. The problem gives us a diagram, which is the key to everything. In geometry, diagrams are your best friends. They visually represent the problem and help you see the relationships between different parts. The question states, "Pada gambar di bawah, $LA = 10\text{ cm}$ dan $AC = 12\text{ cm}$ . Dengan demikian panjang $DE$ sama dengan?" Which basically translates to, "In the figure below, $LA = 10\text{ cm}$ and $AC = 12\text{ cm}$ . Therefore, the length of $DE$ is equal to what?"

So, what do we have? We've got a diagram with some line segments and some lengths given to us. We need to use these pieces of information to find the length of $DE$ . The most important part is to focus on the givens: the lengths of $LA$ and $AC$ . With these, we have a starting point and some information that will help us find the length of the unknown side $DE$ . The question is, how do we get from those known lengths to $DE$ ? That's where similar triangles and proportions come into play. This is a key principle in geometry, so make sure you understand it!

As we work through this problem, remember to take your time and break it down into smaller, more manageable steps. Don't be afraid to draw your own diagrams, label the sides, and mark the angles. This will help you visualize the problem and stay organized. The more you practice, the more confident you'll become in solving these types of problems. And trust me, it's a great feeling to finally crack a geometry problem! So, let's keep going and find the solution. Remember to always double-check your work and make sure your answer makes sense in the context of the problem.

Identifying Key Concepts: Similar Triangles and Proportions

Okay, guys, here comes the fun part! The secret sauce to solving this problem lies in understanding similar triangles. What are they? Well, similar triangles are triangles that have the same shape but can be different sizes. This means their corresponding angles are equal, and their corresponding sides are in proportion. This is super important because it provides us with a relationship that we can use to find the unknown length $DE$ .

How do we spot similar triangles in our diagram? Look for triangles that have the same angles. If two angles of one triangle are equal to two angles of another triangle, then the third angles must also be equal, and the triangles are similar. Once you've identified the similar triangles, you can set up proportions using the corresponding sides. A proportion is simply an equation that states that two ratios are equal. In our case, we'll use proportions to relate the sides of the similar triangles and solve for $DE$ . Remember, the order in which you set up your proportions matters. Make sure you match the corresponding sides correctly. For example, if you're comparing the sides of two triangles, make sure you compare the longest side of one triangle to the longest side of the other, the shortest side to the shortest side, and so on.

This is a fundamental concept in geometry, so understanding how similar triangles work is critical. If you are ever stuck on how to proceed, review the properties of similar triangles. Remember, the ratio of corresponding sides is always the same. So, if we know the lengths of some sides in one triangle and the length of a corresponding side in the other triangle, we can use proportions to find the length of the unknown side. Proportions are really powerful tools! In short, the concept of similar triangles is the key to unlock the problem and calculate the unknown length of $DE$ . So, keep the idea of proportions in mind, it will be the most useful method to solve the equation. Always review the concepts to feel secure in your solving steps.

Setting Up the Proportion and Solving for DE

Alright, time to get our hands dirty and actually solve for $DE$ ! Now that we know about similar triangles and proportions, we can use them to find the missing length. First, we need to figure out which triangles are similar and then set up our proportion.

Looking at the diagram, we can assume that we're dealing with similar triangles (although the problem doesn't explicitly state it, it's a common assumption in these types of problems). We can infer that there are two triangles in the figure: a smaller one and a larger one. We'll set up our proportion using the ratio of corresponding sides. Let's say that $LA$ corresponds to $DE$ , and $AC$ corresponds to some other side of the larger triangle (we can work that out later). We know that $LA = 10\text{ cm}$ and $AC = 12\text{ cm}$ . We can represent our proportion like this: $\frac{LA}{DE} = \frac{AC}{?}$ Here we are looking for the length of $DE$ ! To solve for $DE$ , we'll need to figure out the corresponding side for $AC$ . Once we identify the corresponding sides, we will substitute the given values and solve the equation. The key is to make sure we're comparing the correct sides of the similar triangles. Then, we can use cross-multiplication (or other algebraic manipulations) to isolate $DE$ and find its length. Remember to always double-check your work and make sure your answer makes sense in the context of the problem. It's easy to make a small mistake when working with proportions, so take your time and be careful. After we set up the proportion, we'll solve for $DE$ and arrive at our final answer. That final answer will be one of the multiple-choice options, which will help us check our work to ensure we are right.

Step-by-Step Solution

Let's break down the solution step-by-step to make sure everything is crystal clear. This is how to get the final answer. Ready? Here we go:

Identify Similar Triangles:
- In the given diagram, we assume that the two triangles are similar, which means their corresponding angles are equal. This is the foundation of our solution.
Set Up the Proportion:
- We can set up a proportion using the corresponding sides of the similar triangles. Let's assume that $LA$ corresponds to $DE$ , and $AC$ corresponds to $AE$ (where $E$ is a point on the line segment).
- The proportion will be: $\frac{LA}{DE} = \frac{AC}{AE}$
Find the missing value of AE:
- AE can be found by adding AC and CE to find the total value, which would be 12 + something. Without the figure, we cannot determine the exact value of CE, which means we cannot determine the exact value of AE. We can assume that if LA corresponds to DE, then AC corresponds to AE. Therefore, $\frac{LA}{DE} = \frac{AC}{AE}$ turns into $\frac{10}{DE} = \frac{12}{AE}$ .
Solve for DE:
- Rearrange the proportion equation: $DE = \frac{LA \times AE}{AC}$ .
- Since we don't have enough information to find AE, we must assume that the information is in the figure. Let's assume the length of AE is double the length of AC. This means AE = 24. We can then insert it to the equation: $DE = \frac{10 \times 24}{12}$ .
- Therefore, $DE = 20$ . However, this is not in the options.
Re-evaluate based on the provided options:
- Since the original plan did not work, let's rearrange the proportion in a way to match the options. We will rearrange the proportion from step 2. We can try setting up a proportion like this: $\frac{LA}{AC} = \frac{10}{12}$ and setting it up with the options. Since we don't know the exact value of AE, and the value of DE, we need to find a new pattern.
- A. $16\text{ cm}$ : $\frac{10}{12} = \frac{DE}{AE}$ . Solving for AE, the value needs to be 19.2, which is not possible.
- B. $18\text{ cm}$ : $\frac{10}{12} = \frac{DE}{AE}$ . Solving for AE, the value needs to be 21.6, which is not possible.
- C. $21\text{ cm}$ : $\frac{10}{12} = \frac{DE}{AE}$ . Solving for AE, the value needs to be 25.2, which is not possible.
- D. $22\text{ cm}$ : $\frac{10}{12} = \frac{DE}{AE}$ . Solving for AE, the value needs to be 26.4, which is not possible.
- E. $24\text{ cm}$ : $\frac{10}{12} = \frac{DE}{AE}$ . Solving for AE, the value needs to be 28.8, which is not possible.
Final answer:
- The answer choices are most likely wrong, or the figure is missing some information. Given this, none of the answers are possible with our current information, so we cannot determine the correct answer.

Conclusion: The Final Answer and Key Takeaways

Alright, guys, we've walked through the problem, and while we faced some roadblocks due to the missing information, we did learn and exercise the principles of solving the problems. The length of $DE$ cannot be precisely determined without more data, but that's alright! The most important thing is that we understand the process. We identified similar triangles, set up a proportion, and worked through the steps to solve for the unknown. We also learned how to double-check our work and make sure our answer makes sense.

So, what are the key takeaways from this problem? First and foremost, remember the power of similar triangles and proportions. They are your best friends when it comes to solving geometry problems. Second, always take your time and break the problem down into smaller, more manageable steps. Draw diagrams, label sides, and mark angles. Third, don't be afraid to experiment and try different approaches. If one method doesn't work, try another! The more you practice, the better you'll become at recognizing patterns and applying the right formulas. Keep practicing, and you'll be a geometry pro in no time! Keep in mind that for this problem, we were missing some information, but that doesn't mean we cannot go through the process of solving it. You can do it!