Finding 'p' In An Isosceles Triangle: A Math Problem

Hey guys! Let's dive into a fun math problem involving an isosceles triangle. We're given the coordinates of three points, A, B, and C, and the condition that the triangle ABC is isosceles, meaning two sides have equal lengths. Our mission? To find the value of 'p' within the coordinates of point B. Sounds like a blast, right? Let's break it down step by step to make sure we understand it perfectly. It's all about using some basic geometry and a little bit of algebra to crack this code. This problem will not only help you understand the concept but also give you confidence in solving similar problems in the future. So, grab your pencils, and let's get started. We'll be using the distance formula, a crucial tool for calculating the length of line segments in a coordinate system. This formula is derived from the Pythagorean theorem, which we all know and love, and it's super handy for this type of problem. Remember, practice makes perfect, so don't worry if it seems a bit tricky at first; we'll walk through it together.

First, let's look at the given information. We have an isosceles triangle ABC, where: A = (11, 8, 9), B = (-1, 2p, 3), and C = (3, -2, -9). Also, we know that the length of the side AB is equal to the length of the side BC ( |AB| = |BC| ). This is the key piece of information we'll use to solve for 'p'. In an isosceles triangle, at least two sides are equal in length. Since we know AB = BC, we can use the distance formula to set up an equation and solve for 'p'. The distance formula will help us find the length of each side. So, let's get into the distance formula and apply it to each side to help us understand. This is a fundamental concept in coordinate geometry, so grasping it is super important. We'll calculate the distance between points A and B, and then the distance between points B and C. By setting these two distances equal to each other (because the triangle is isosceles and AB = BC), we can solve for 'p'. The algebra part will be relatively simple, but the key is to apply the distance formula correctly. Remember, precision is super important in this type of problem, so we'll be super careful with our calculations. Always double-check your work to avoid silly mistakes. Ready to see the magic happen? Let's go!

Understanding the Distance Formula

Alright, before we get our hands dirty with the calculations, let's quickly recap the distance formula. This formula helps us find the distance between two points in a 3D space, which is exactly what we need to find the lengths of the sides of our triangle. The distance formula is essentially an extension of the Pythagorean theorem to three dimensions. If we have two points, P(x1, y1, z1) and Q(x2, y2, z2), the distance between them is given by: √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²). See? Not too scary, right? It's just about subtracting the coordinates, squaring the differences, adding them up, and taking the square root. The distance formula is our best friend in this problem. It allows us to calculate the lengths of the sides of the triangle. Understanding this formula is the key to solving the problem. The distance formula is essential for calculating the length of line segments in coordinate geometry. This formula helps us understand the relationship between the coordinates of points and the distances between them. Remember, the square root of the sum of the squares of the differences in the x, y, and z coordinates gives you the distance. This is the foundation upon which we'll build our solution. It's super important to memorize this formula or at least be very familiar with it. This way, we can quickly calculate the lengths of the sides of the triangle and solve for 'p'. Let's move on and use this formula to calculate the lengths of AB and BC.

Now, let's apply the distance formula. First, let's find the length of AB. Using the coordinates of A(11, 8, 9) and B(-1, 2p, 3), we get: AB = √((-1 - 11)² + (2p - 8)² + (3 - 9)²). Simplifying this, we get: AB = √((-12)² + (2p - 8)² + (-6)²), which further simplifies to AB = √(144 + (2p - 8)² + 36). Now, let's simplify a bit more: AB = √(180 + (2p - 8)²). Great! We've found the length of AB in terms of 'p'. Next, we'll find the length of BC. Using the coordinates of B(-1, 2p, 3) and C(3, -2, -9), we get: BC = √((3 - (-1))² + (-2 - 2p)² + (-9 - 3)²). Let's simplify that: BC = √((4)² + (-2 - 2p)² + (-12)²), which becomes BC = √(16 + (-2 - 2p)² + 144). Further simplifying, we get: BC = √(160 + (-2 - 2p)²). Now, let's move on to the next step, where we equate AB and BC and solve for 'p'. We are now ready to tackle the main part of the problem: finding the value of 'p' by equating the lengths of AB and BC. Remember, since the triangle is isosceles and AB = BC, we can set up an equation and solve for 'p'. The algebra may seem a bit tricky, but with careful calculations, we'll get there. Always double-check your work to avoid silly mistakes. So, are you guys ready to solve for 'p'? Here we go!

Solving for 'p'

Now that we have the expressions for the lengths of AB and BC, we can set them equal to each other because we know that |AB| = |BC|. So, we have: √(180 + (2p - 8)²) = √(160 + (-2 - 2p)²). To get rid of the square roots, let's square both sides of the equation. This gives us: 180 + (2p - 8)² = 160 + (-2 - 2p)². Expanding the squares, we get: 180 + (4p² - 32p + 64) = 160 + (4 + 8p + 4p²). Now, let's simplify and solve for 'p'. Combine like terms: 180 + 4p² - 32p + 64 = 160 + 4 + 8p + 4p². This simplifies to: 244 + 4p² - 32p = 164 + 8p + 4p². Subtracting 4p² from both sides, we get: 244 - 32p = 164 + 8p. Now, let's move all the 'p' terms to one side and constants to the other. Adding 32p to both sides and subtracting 164 from both sides, we get: 244 - 164 = 8p + 32p. This simplifies to: 80 = 40p. Finally, dividing both sides by 40, we get: p = 2. Great job, everyone! We've successfully found the value of 'p'. This part is where we put our equation-solving skills to the test. With each step, we've gotten closer to solving for 'p'. So, we've successfully found the value of 'p', and it turns out to be 2. This is a common method for solving problems that involve distances and coordinates. With the value of 'p', we can now know the exact coordinates of point B.

Now, let's double-check our answer and ensure it makes sense. If p = 2, then the coordinates of B are (-1, 4, 3). With these coordinates, we can recalculate the lengths of AB and BC to verify that they are indeed equal. This is always a great practice to make sure you haven't made any mistakes. Let's recap what we've done. We started with an isosceles triangle, used the distance formula to find the lengths of the sides, set the lengths of AB and BC equal to each other, and solved the resulting equation for 'p'. It’s really awesome to see how geometry and algebra work together to solve a problem. It might seem tricky at first, but with a good understanding of the distance formula and some algebraic skills, these problems become much more manageable. You guys should feel really proud of yourselves. You've successfully solved for 'p' in an isosceles triangle problem! Keep practicing and you'll get even better. Remember to always double-check your calculations and keep practicing. You've got this!

Conclusion and Answer

So, after all that hard work, we've found that p = 2. This means the correct answer from the given options is B. 2. We've successfully navigated through the distance formula, squared equations, and algebraic manipulations to arrive at the solution. High five, everyone! You've successfully solved an isosceles triangle problem. Remember, the key to solving these kinds of problems is a solid understanding of the distance formula and the ability to manipulate algebraic equations. Keep practicing, and you'll become a pro in no time! Keep in mind that math problems like this are not just about finding the right answer; they are also about developing critical thinking and problem-solving skills. Remember that every problem you solve makes you better equipped to tackle the next one. So keep up the amazing work! If you encounter similar problems in the future, you'll be well-prepared to handle them. You’ve done a great job today. Keep up the awesome work!