Finding Side BC In Triangle ABC A Step-by-Step Solution
Alright, let's dive into this geometry problem together! We've got a triangle ABC, and we're given some information about its angles and sides. Our mission, should we choose to accept it, is to find the length of side BC. Don't worry, it's not as daunting as it seems! We'll break it down step by step, just like we're solving a puzzle with our favorite math tools.
Understanding the Problem
First, let's recap what we know. We're dealing with a triangle ABC, and we have the following clues:
- cos(B+C) = 9/40
- AC = 10 cm
- AB = 8 cm
Our goal is to determine the length of BC. To tackle this, we'll need to pull some trigonometric magic out of our hats and remember a few key concepts about triangles. Think of it as gathering our ingredients before we start cooking up a solution. Trigonometry will be our main ingredient here, helping us relate angles and sides.
Key Trigonometric Concepts
Before we jump into the calculations, let’s quickly refresh some fundamental trigonometric principles that will be pivotal in solving this problem. These concepts are the backbone of our approach, ensuring we have a solid foundation for every step we take.
The Cosine Rule: A Cornerstone for Solving Triangles
The cosine rule is a gem when it comes to relating the sides and angles of a triangle. It states that for any triangle with sides a, b, and c, and angles A, B, and C opposite those sides, the following relationship holds:
This rule is particularly useful when we know two sides and the included angle (the angle between them) or when we know all three sides and want to find an angle. In our case, we'll be leveraging this rule to connect the sides AB, AC, and BC, and the angle A.
The Angle Sum Property: Triangles Always Add Up!
One of the most basic yet crucial properties of triangles is that the sum of the angles in any triangle is always 180 degrees (or π radians). Mathematically, this is expressed as:
This property is invaluable because it allows us to find a missing angle if we know the other two. In our problem, this property will help us relate the angle A to the given cos(B+C), providing a vital link in our solution chain.
Cosine of Supplementary Angles: Navigating the Trigonometric Landscape
Supplementary angles are angles that add up to 180 degrees. There's a neat relationship between the cosines of supplementary angles that we can use. If two angles, say θ and φ, are supplementary (i.e., θ + φ = 180°), then:
This property arises from the symmetry of the cosine function and is incredibly useful when we're dealing with expressions like cos(B+C). By understanding this relationship, we can manipulate trigonometric expressions more effectively, paving the way for a cleaner solution.
Cracking the Code: Step-by-Step Solution
Now that we've got our concepts locked and loaded, let's start piecing together the solution. Our first step involves using the angle sum property to relate cos(B+C) to cos(A). This is where the magic begins!
Step 1: Finding cos(A)
We know that A + B + C = 180°. So, we can express A as:
A = 180° - (B + C)
Now, let's take the cosine of both sides:
cos(A) = cos(180° - (B + C))
Using the property of supplementary angles, we know that cos(180° - x) = -cos(x). Therefore:
cos(A) = -cos(B + C)
We're given that cos(B + C) = 9/40, so:
cos(A) = -9/40
Awesome! We've found cos(A). This is a crucial piece of the puzzle. Now, we can move on to the next step, where we'll employ the cosine rule to bring the sides into the equation.
Step 2: Applying the Cosine Rule
The cosine rule states:
a² = b² + c² - 2bc * cos(A)
In our triangle, let a = BC (what we want to find), b = AC = 10 cm, and c = AB = 8 cm. Plugging these values and the value of cos(A) into the cosine rule, we get:
BC² = 10² + 8² - 2 * 10 * 8 * (-9/40)
Now, let's simplify this expression. It's just a matter of arithmetic from here!
Step 3: Crunching the Numbers
Let's break down the calculation step by step:
BC² = 100 + 64 - 160 * (-9/40)
BC² = 164 + (160 * 9) / 40
BC² = 164 + (16 * 9) / 4
BC² = 164 + 4 * 9
BC² = 164 + 36
BC² = 200
So, BC² = 200. To find BC, we simply take the square root of both sides.
Step 4: Finding BC
BC = √200
We can simplify √200 by factoring out the largest perfect square:
BC = √(100 * 2)
BC = √100 * √2
BC = 10√2 cm
Ta-da! We've found the length of side BC. It's 10√2 cm. Pat yourself on the back, guys – you've earned it!
The Final Answer
The length of side BC is 10√2 cm. So, the correct answer is (c). We've successfully navigated the trigonometric terrain and emerged victorious with our solution. Wasn't that fun?
Why This Solution Works: A Recap
Let's take a moment to appreciate the journey we've been on and recap why our solution works so well. We started with a seemingly complex problem and broke it down into manageable steps, using key trigonometric principles as our guide. It's like following a map to a hidden treasure, with each step bringing us closer to our goal.
The Power of Trigonometric Relationships
At the heart of our solution lies the power of trigonometric relationships. By understanding how angles and sides relate to each other in a triangle, we were able to connect the given information to what we needed to find. The cosine rule, in particular, was instrumental in bridging the gap between the known sides and angles and the unknown side BC.
The Importance of Angle Sum Property
The angle sum property, a fundamental characteristic of triangles, played a crucial role in our solution. It allowed us to relate the given cos(B+C) to cos(A), providing a vital link in our chain of reasoning. This property underscores the importance of remembering basic geometric principles, as they often hold the key to unlocking more complex problems.
Breaking Down the Problem
One of the most effective strategies in problem-solving is breaking down the problem into smaller, more manageable parts. That's exactly what we did here. By focusing on one step at a time – finding cos(A), applying the cosine rule, simplifying the expression – we avoided feeling overwhelmed and made the solution process much smoother. It's like climbing a mountain one step at a time, rather than trying to scale it in one giant leap.
The Elegance of Mathematical Reasoning
There's an inherent elegance in mathematical reasoning. Each step flows logically from the previous one, building a solid and coherent solution. Our solution exemplifies this elegance, showcasing how a combination of trigonometric principles, algebraic manipulation, and careful calculation can lead us to a clear and concise answer. It's like a well-composed symphony, where each note plays its part in creating a harmonious whole.
Practice Makes Perfect: Similar Problems to Try
Now that we've conquered this problem, let's keep the momentum going! The best way to solidify your understanding and hone your problem-solving skills is to practice. Here are a few similar problems you can try:
- In triangle PQR, given cos(Q+R) = -5/13, PQ = 7 cm, and PR = 9 cm, find the length of side QR.
- In triangle XYZ, if XY = 12 cm, YZ = 15 cm, and cos(Y) = 1/4, find the length of side XZ.
- In triangle ABC, given AB = 6 cm, AC = 8 cm, and BC = 10 cm, find the value of cos(A).
Try tackling these problems using the same strategies we employed in the solution above. Remember to break down the problem, identify the key concepts, and work through each step systematically. The more you practice, the more confident and proficient you'll become in solving geometry problems.
Conclusion: Geometry Adventures Await!
Well, guys, we've reached the end of our geometric adventure for today. We successfully navigated the world of triangles, cosines, and square roots, and emerged with a satisfying solution. Remember, geometry is not just about formulas and theorems; it's about spatial reasoning, logical thinking, and the thrill of uncovering hidden relationships. So, keep exploring, keep practicing, and keep enjoying the beauty of mathematics. Who knows what geometric wonders we'll discover next time?
