How To Find The Third Side Of A Triangle When Two Sides And The Perimeter Are Known

Jul 28, 2025 by ADMIN 84 views

Solving Triangle Side Length Problems: A Step-by-Step Guide

Hey guys! Geometry can be super fun, especially when we're dealing with shapes like triangles. Today, we're going to break down a classic triangle problem step-by-step, so you can conquer similar questions with confidence. Let's dive in!

Understanding the Triangle Problem

Our problem involves a triangle where we know the lengths of two sides and the total length of all sides combined (the perimeter). The challenge is to find the length of the third side. This is a common type of geometry problem, and mastering it will help you tackle more complex scenarios later on. Before we jump into calculations, let's quickly review some basic triangle facts.

Triangles, as you know, are three-sided shapes, and the sum of all three sides gives us the perimeter. In our case, we know two sides and the perimeter, which means we can use simple math to find the missing side. Think of it like this: if you have a rope of a certain length (perimeter) and you've already used up some of it for two sides, the remaining rope length will be the third side. The key here is understanding the relationship between the sides and the perimeter. We can express this relationship with a simple formula: Side 1 + Side 2 + Side 3 = Perimeter. This is the foundation of our solution. Now, let's get into the specifics of our problem. We know Side 1 is 5 cm, Side 2 is 9 cm, and the Perimeter is 24 cm. Our mission is to find Side 3. We'll use the formula we just discussed and a bit of algebra to uncover the answer. Remember, geometry problems are often like puzzles – you just need to fit the pieces together correctly! We'll carefully substitute the known values into the formula and then isolate the unknown variable (Side 3) to solve for it. Don't worry, it's easier than it sounds! So, grab your pencils and let's get started on solving this triangle mystery.

Step-by-Step Solution

Okay, so let's break down this triangle problem and find the missing side. Remember, we're trying to find the length of the third side of a triangle, given that one side is 5 cm, another is 9 cm, and the total perimeter is 24 cm.

Here’s how we'll do it:

Recall the Perimeter Formula: The perimeter of a triangle is simply the sum of the lengths of its three sides. We can write this as: Perimeter = Side 1 + Side 2 + Side 3
Plug in the Known Values: We know the perimeter (24 cm), Side 1 (5 cm), and Side 2 (9 cm). Let’s substitute these values into our formula: 24 cm = 5 cm + 9 cm + Side 3. This step is crucial because it translates the word problem into a mathematical equation. By substituting the known values, we've transformed the problem into a form that we can easily solve. It's like filling in the blanks in a sentence – we have most of the information, and now we just need to find the missing piece. This process of substitution is a fundamental technique in algebra and geometry, and it's used extensively in solving various types of problems. It allows us to represent real-world situations using mathematical symbols and equations, making it easier to analyze and find solutions. So, make sure you're comfortable with this step, as it's a building block for more advanced problem-solving.
Simplify the Equation: Now, let's simplify the right side of the equation by adding 5 cm and 9 cm: 24 cm = 14 cm + Side 3. Simplifying equations is like tidying up a room – it makes everything clearer and easier to work with. In this case, we're combining the known quantities on one side of the equation to isolate the unknown variable (Side 3). This process is based on the basic principles of arithmetic and algebra, where we can perform operations on both sides of an equation without changing its balance. Think of an equation as a balanced scale – if you add or subtract the same weight from both sides, the scale remains balanced. Similarly, in our equation, we're combining the numerical terms to simplify the expression and bring us closer to the solution. This step is not only about making the equation look cleaner; it's also about making the relationship between the variables more apparent. By simplifying, we can clearly see how Side 3 relates to the other known values, which is essential for solving the problem. So, let's keep simplifying and get closer to finding the length of that elusive third side!
Isolate Side 3: To find the length of Side 3, we need to isolate it on one side of the equation. We can do this by subtracting 14 cm from both sides: 24 cm - 14 cm = Side 3. Isolating a variable in an equation is like finding the hidden treasure in a puzzle. It's the key step that reveals the value we're looking for. In this case, we want to find the length of Side 3, so we need to get it all by itself on one side of the equation. To do this, we use the principle of inverse operations. Since 14 cm is being added to Side 3, we perform the opposite operation, which is subtraction, to "undo" the addition. But remember, like a balanced scale, we need to do the same thing to both sides of the equation to maintain the equality. So, we subtract 14 cm from both sides, which cancels out the 14 cm on the right side, leaving Side 3 isolated. This step is a fundamental concept in algebra, and it's used in solving a wide range of equations. By mastering this technique, you'll be well-equipped to tackle more complex problems in math and beyond. So, let's complete the subtraction and uncover the value of Side 3!
Calculate Side 3: Now, perform the subtraction: 10 cm = Side 3. Finally, we've arrived at the answer! By performing the subtraction, we've successfully calculated the length of Side 3. This step is the culmination of all our previous efforts, where we applied our understanding of the perimeter formula, substituted the known values, simplified the equation, and isolated the unknown variable. It's like reaching the summit of a mountain after a challenging climb – we've reached our goal! The result, 10 cm, tells us the exact length of the third side of the triangle. This completes our solution and answers the original question posed in the problem. But it's not just about getting the right answer; it's also about understanding the process and the underlying concepts. By following these steps, you've not only solved this particular problem but also gained valuable problem-solving skills that can be applied to other geometry problems and beyond. So, take a moment to appreciate your accomplishment and the knowledge you've gained!

The Answer

So, the length of the third side of the triangle is 10 cm. Awesome job! You've successfully solved the problem by applying the perimeter formula and using some basic algebraic techniques. Always remember to double-check your work and make sure your answer makes sense in the context of the problem.

Practice Makes Perfect

Geometry, like any math topic, gets easier with practice. Try solving similar problems with different side lengths and perimeters. You can even challenge yourself by working with different types of triangles, like isosceles or equilateral triangles. The more you practice, the more confident you'll become in your problem-solving skills. Remember, the key to success in geometry is understanding the fundamental concepts and applying them consistently. So, keep practicing, keep exploring, and keep having fun with shapes!

Real-World Applications

Understanding triangles and their properties isn't just about solving textbook problems. It has practical applications in various fields, such as:

Architecture: Architects use triangles extensively in building designs for their structural stability.
Engineering: Engineers rely on triangular shapes in bridge construction and other structural designs.
Navigation: Triangles are fundamental in trigonometry, which is used in navigation systems.
Art and Design: Artists and designers use triangles to create visually appealing compositions.

So, the knowledge you've gained today is not just theoretical; it's something you can apply in the real world. By understanding the properties of triangles, you're opening doors to a wide range of possibilities and applications. It's like learning a new language – the more you understand it, the more you can communicate and express yourself in different ways. So, keep exploring the world of geometry and discover the many ways it shapes our world!

Conclusion

Great work, everyone! We've successfully tackled a triangle problem and learned a valuable problem-solving strategy. Remember, geometry is all about understanding shapes, their properties, and how they relate to each other. By practicing regularly and applying the steps we discussed today, you'll be well on your way to mastering geometry. Keep up the amazing work!