Find FH Length: Line Segment Geometry Problem

Hey guys! Let's dive into a fun geometry problem involving line segments. We've got points F, G, and H on a line, with G sitting snugly between F and H. We're given some expressions for the lengths of the segments FG, GH, and FH, and our mission, should we choose to accept it, is to find the actual length of FH. Don't worry, it's not as daunting as it sounds! We'll break it down step by step, making sure everyone's on board. Think of it as a puzzle – we've got the pieces, now we just need to fit them together. Geometry, at its heart, is all about relationships, and this problem beautifully illustrates the relationship between the parts and the whole of a line segment. So, buckle up, grab your thinking caps, and let's get started! We're going to explore how the lengths of segments relate to each other when they lie on the same line. This is a fundamental concept in geometry, and mastering it will open doors to solving more complex problems down the road. We'll use a bit of algebra, but don't let that scare you. We'll take it slow and steady, explaining each step as we go. By the end of this article, you'll not only know how to solve this particular problem but also have a solid understanding of the underlying principles. This knowledge will empower you to tackle similar challenges with confidence and ease. Remember, mathematics is a journey, not a destination. It's about building understanding step by step, and we're here to guide you on that journey. So, let's embark on this adventure together and unravel the mystery of line segments!

Setting Up the Problem: The Segment Addition Postulate

In geometry problems like this, the key concept we need to remember is the Segment Addition Postulate. What's that, you ask? Simply put, it states that if you have a point (in our case, G) that lies between two other points (F and H) on a line segment, then the sum of the lengths of the two smaller segments (FG and GH) will equal the length of the entire segment (FH). Make sense? It's like saying if you have a small piece of rope and another small piece, and you tie them together, the total length is the sum of the lengths of the individual pieces. Visually, you can imagine the line segment FH being split into two parts by the point G. The Segment Addition Postulate provides the crucial link between these parts and the whole. It's a foundational principle that allows us to translate the geometric relationship into an algebraic equation. This is where the beauty of mathematics shines – we can use algebra to represent and solve geometric problems. Now, let's translate the given information into a mathematical equation using this postulate. We know that FG + GH = FH. We also know the expressions for each of these lengths: FG = 8x - 12, GH = 7x, and FH = 13x + 10. By substituting these expressions into the equation, we'll have an algebraic equation in terms of x. Solving for x will be the next step in our quest to find the length of FH. This process of translating geometric information into algebraic equations is a powerful problem-solving technique that is widely used in mathematics. It allows us to leverage the tools of algebra to solve geometric problems and vice versa. So, let's take a deep breath and move on to the next step: setting up the equation.

Forming the Equation: Plugging in the Values

Alright, let's get down to the nitty-gritty. We're going to take those expressions we have for the lengths of the segments and plug them into our equation from the Segment Addition Postulate: FG + GH = FH. Remember, FG is given as 8x - 12, GH is 7x, and FH is 13x + 10. So, our equation becomes: (8x - 12) + (7x) = (13x + 10). See? Not so scary! We've just replaced the segment names with their corresponding algebraic expressions. Now we have a good old-fashioned algebraic equation to solve. This is a crucial step because it allows us to use the power of algebra to find the value of x. Once we know x, we can then substitute it back into the expression for FH to find its length. Think of this equation as a bridge connecting the geometry of the line segment to the algebra we're about to use. It's the key that unlocks the solution to our problem. The equation represents the relationship between the different segments of the line and allows us to use algebraic manipulation to find the unknown value, x. Setting up the equation correctly is paramount. A mistake here will throw off the entire solution. So, double-check your substitutions and make sure everything is in its right place. We're now poised to solve this equation, and that's what we'll tackle in the next section. We'll use the familiar rules of algebra to isolate x and find its value. So, keep your pencils sharp and your minds focused, and let's move on to the next step in our journey!

Solving for x: Unleashing Our Algebra Skills

Okay, guys, it's algebra time! We've got our equation: (8x - 12) + (7x) = (13x + 10). The first thing we need to do is simplify both sides of the equation. On the left side, we can combine the 'x' terms: 8x + 7x = 15x. So, the left side becomes 15x - 12. Now our equation looks like this: 15x - 12 = 13x + 10. Much cleaner, right? The next step is to get all the 'x' terms on one side of the equation and all the constant terms on the other side. Let's subtract 13x from both sides: 15x - 13x - 12 = 13x - 13x + 10. This simplifies to 2x - 12 = 10. Now, let's add 12 to both sides: 2x - 12 + 12 = 10 + 12. This gives us 2x = 22. Finally, to isolate x, we divide both sides by 2: (2x) / 2 = 22 / 2. And there we have it: x = 11. Hooray! We've found the value of x. But remember, we're not done yet. We were asked to find the length of FH, not the value of x. But finding x was a crucial step in getting there. Now that we know x, we can substitute it back into the expression for FH to find its actual length. Solving for x is a fundamental skill in algebra, and it's used extensively in various mathematical contexts. The steps we followed – simplifying, combining like terms, and isolating the variable – are the bread and butter of algebraic problem-solving. So, make sure you're comfortable with these techniques. Now, with x = 11 in our pocket, let's move on to the final stage of our journey: finding the length of FH.

Finding FH: The Final Substitution

We're in the home stretch now! We know that x = 11, and we know that FH = 13x + 10. So, to find the length of FH, all we need to do is substitute 11 for x in the expression 13x + 10. This gives us FH = 13(11) + 10. Now it's just a matter of arithmetic. 13 multiplied by 11 is 143. So, FH = 143 + 10. And finally, 143 + 10 = 153. Therefore, the length of FH is 153. Woohoo! We did it! We've successfully found the length of the line segment FH. This final substitution is the culmination of all our hard work. We used the Segment Addition Postulate to set up an equation, we used our algebra skills to solve for x, and now we're using that value of x to find the answer to our original question. This process highlights the interconnectedness of different mathematical concepts. Geometry provides the framework, algebra provides the tools, and arithmetic brings it all home. It's important to remember to always go back to the original question and make sure you've answered it fully. We found x, but the question asked for the length of FH. So, the substitution step was crucial to providing the complete answer. Now, let's take a moment to recap our journey and appreciate the steps we took to arrive at the solution.

Wrapping Up: Our Journey to Finding FH

Let's take a moment to pat ourselves on the back. We started with a geometry problem involving line segments, and we successfully navigated our way to the solution. We were given that G is between F and H on a line segment, and we had expressions for the lengths of FG, GH, and FH in terms of x. Our goal was to find the length of FH. We began by understanding the Segment Addition Postulate, which states that FG + GH = FH. This was the foundation upon which we built our solution. We then substituted the given expressions into the equation, resulting in (8x - 12) + (7x) = (13x + 10). This transformed our geometric problem into an algebraic one. Next, we unleashed our algebra skills to solve for x. We simplified the equation, combined like terms, and isolated x, ultimately finding that x = 11. With the value of x in hand, we substituted it back into the expression for FH, which was 13x + 10. This gave us FH = 13(11) + 10. Finally, we performed the arithmetic to find that FH = 153. We've not only found the answer to this specific problem, but we've also reinforced our understanding of the Segment Addition Postulate and honed our algebraic problem-solving skills. Remember, mathematics is about building a strong foundation of concepts and skills that you can apply to a wide range of problems. So, keep practicing, keep exploring, and keep challenging yourself. And most importantly, have fun with it! Now you know how to tackle similar line segment problems with confidence. Great job, everyone!** The length of FH is 153 **. We started by remembering the ** Segment Addition Postulate *** which states that the sum of the lengths of the two smaller segments (FG and GH) will be equal to the length of the entire segment (FH) . This simple but powerful principle is the key to unlocking this problem and many others in geometry. This allows us to translate the given information into a mathematical equation and solve for the unknown. This translation from geometric representation to algebraic equation is a core skill in mathematics and is applicable in various fields. We carefully substituted the given expressions into the equation, making sure each term was placed correctly. Accuracy in this step is paramount, as any error will propagate through the rest of the solution. We simplified the equation by combining like terms and isolating the variable * x *. These are fundamental algebraic techniques that are used extensively in solving equations. Once we found the value of * x *, we weren't done yet! We remembered the original question, which asked for the length of FH, not the value of * x *. This highlights the importance of always keeping the end goal in mind. We substituted the value of * x * back into the expression for FH and performed the final calculation. This gave us the answer we were looking for: FH = 153 . We reviewed our steps to ensure accuracy and completeness. This reflective practice is crucial for learning and retaining mathematical concepts. By understanding not just the steps but also the reasoning behind them, we deepen our understanding and improve our problem-solving abilities. So, next time you encounter a similar problem, remember the Segment Addition Postulate, the power of algebra, and the importance of careful calculation. You've got this! This problem serves as a great example of how geometry and algebra work together to solve real-world problems. The logical progression of steps, from understanding the geometric principle to applying algebraic techniques, is a hallmark of mathematical problem-solving. Remember, practice makes perfect. The more problems you solve, the more comfortable you'll become with these concepts and the more confident you'll be in your abilities. So, keep exploring, keep learning, and keep having fun with math! If we have * points F *, * G *, and * H * on a line, with * G * between * F * and * H *, we know that * FG * + * GH * = * FH *. This is because the sum of the parts equals the whole. We use this postulate to set up an equation based on the given information. We have * FG * = 8 * x * - 12, * GH * = 7 * x *, and * FH * = 13 * x * + 10. Substituting these expressions into the equation, we get 8 * x * - 12 + 7 * x * = 13 * x * + 10. This step translates the geometric relationship into an algebraic one, allowing us to use algebraic techniques to solve for * x *. We can simplify the equation by combining like terms on the left side: (8 * x * + 7 * x *) - 12 = 15 * x * - 12. So, the equation becomes 15 * x * - 12 = 13 * x * + 10. Now, we want to isolate * x * on one side of the equation. Subtracting 13 * x * from both sides gives us 15 * x * - 13 * x * - 12 = 13 * x * - 13 * x * + 10, which simplifies to 2 * x * - 12 = 10. Next, we add 12 to both sides of the equation: 2 * x * - 12 + 12 = 10 + 12, which simplifies to 2 * x * = 22. Finally, we divide both sides by 2 to solve for * x *: (2 * x *) / 2 = 22 / 2, so * x * = 11. Now that we know * x * = 11, we can substitute this value into the expression for * FH *: * FH * = 13 * x * + 10 = 13(11) + 10. We calculate 13 * 11 = 143, so * FH * = 143 + 10 = 153. Therefore, the length of * FH * is 153. This final calculation provides the answer to the problem, completing the solution. We verified the answer by reviewing each step of the process to ensure no errors were made. This is a crucial step in problem-solving to ensure accuracy. The problem reinforces the Segment Addition Postulate and algebraic problem-solving skills. These skills are fundamental in geometry and algebra. Solving similar problems will further solidify the understanding of these concepts. This problem illustrates the power of translating geometric problems into algebraic equations. This approach is widely used in mathematics and related fields. The steps we took to solve this problem can be applied to other similar problems. Understanding the process is as important as finding the answer. We can also think about the problem visually to enhance our understanding. Drawing a diagram of the line segment with points * F *, * G *, and * H * can be helpful. This visual representation can aid in grasping the relationships between the segments. The key is to break down complex problems into smaller, manageable steps. This strategy is effective in problem-solving across various domains. With practice, these steps will become more intuitive and you'll be able to solve similar problems more efficiently. Remember, the goal is not just to get the right answer, but to understand the underlying concepts and develop problem-solving skills. This understanding will serve you well in future mathematical endeavors. So, keep practicing, keep exploring, and keep challenging yourself! You've got this!