Polygon Diagonals: Find The Shape With 108 Diagonals
Alright, let's dive into the fascinating world of polygons and their diagonals! This is a classic geometry problem, and we're going to break it down step by step. Our mission is to find the polygon that boasts precisely 108 diagonals. We have four options to consider: a hexagon, an octagon, a decagon, and a dodecagon. To solve this, we'll use the formula for calculating the number of diagonals in a polygon. Buckle up, geometry enthusiasts, because we're about to have some fun!
Understanding Diagonals in Polygons
First, let's make sure we're all on the same page about what a diagonal actually is. In the context of polygons, a diagonal is a line segment that connects two non-adjacent vertices. Basically, it's a line you can draw inside the polygon that isn't one of the sides. For example, in a square, you can draw two diagonals. As the number of sides increases, the number of diagonals shoots up pretty quickly. Think about it: each vertex can connect to every other vertex except itself and the two adjacent ones.
The key to solving this problem is understanding the formula that relates the number of sides of a polygon to the number of diagonals it has. This formula is derived from combinatorics, considering how many ways you can choose two vertices from the polygon and then subtracting the number of sides (since those aren't diagonals). The formula is essential for tackling these types of problems efficiently and accurately.
So, why is this important? Well, diagonals play a crucial role in many areas of geometry and even in practical applications. They help us understand the properties of polygons, analyze their symmetry, and even use them in constructions and designs. Understanding the relationship between the number of sides and the number of diagonals gives us a powerful tool for analyzing shapes.
The Formula for Calculating Diagonals
The number of diagonals (D) in a polygon with n sides is given by the formula:
D = n(n - 3) / 2
This formula works because, from each vertex, you can draw a diagonal to every other vertex except for itself and the two adjacent vertices. That's why we have the (n - 3) part. We then multiply by n to account for each vertex, but we divide by 2 because each diagonal is counted twice (once from each endpoint). This nifty little formula is the key to unlocking our problem.
Let's break down why this formula is so effective. Imagine you're at one corner of the polygon. You can't draw a diagonal to the corner you're standing on, nor can you draw one to the two corners right next to you (because those would just be sides). That leaves n - 3 corners you can draw diagonals to. Now, if you do this for every corner, you'll quickly realize you're counting each diagonal twice (once from each end). Hence, the division by 2. It's elegant, isn't it?
Knowing this formula isn't just useful for this specific problem; it's a fundamental tool in geometry. It allows you to quickly determine the number of diagonals in any polygon, which can be super handy in various mathematical contexts, from simple problem-solving to more complex geometric proofs and constructions. Trust me, having this formula in your back pocket will make your life a lot easier.
Applying the Formula to the Options
Now that we have our formula, let's test each of the given options to see which one has exactly 108 diagonals.
A) Hexagon (n = 6)
D = 6(6 - 3) / 2 = 6(3) / 2 = 18 / 2 = 9
A hexagon has 9 diagonals. So, option A is incorrect.
B) Octagon (n = 8)
D = 8(8 - 3) / 2 = 8(5) / 2 = 40 / 2 = 20
An octagon has 20 diagonals. Thus, option B is also incorrect.
C) Decagon (n = 10)
D = 10(10 - 3) / 2 = 10(7) / 2 = 70 / 2 = 35
A decagon has 35 diagonals. Therefore, option C is incorrect.
D) Dodecagon (n = 12)
D = 12(12 - 3) / 2 = 12(9) / 2 = 108 / 2 = 54
A dodecagon has 54 diagonals. Thus, option D is incorrect.
Oops! It seems there was a slight error in the calculation or the options provided. None of the given polygons have 108 diagonals. Let's find the correct polygon using the formula. We need to solve for n when D = 108.
Finding the Correct Polygon
We have the equation:
108 = n(n - 3) / 2
Multiply both sides by 2:
216 = n(n - 3)
This simplifies to a quadratic equation:
n^2 - 3n - 216 = 0
Now, we need to factor this quadratic equation. We're looking for two numbers that multiply to -216 and add up to -3. Those numbers are -15 and 12.
So, we can rewrite the equation as:
(n - 15)(n + 12) = 0
The possible values for n are 15 and -12. Since the number of sides of a polygon can't be negative, we have n = 15.
Therefore, the polygon with 108 diagonals has 15 sides, which is called a pentadecagon.
Conclusion
In conclusion, after reviewing our calculations and the given options, it appears there was a mistake. None of the provided polygons (hexagon, octagon, decagon, dodecagon) have exactly 108 diagonals. By using the formula D = n(n - 3) / 2 and solving the quadratic equation, we found that the polygon with 108 diagonals is actually a pentadecagon, which has 15 sides. So, the correct answer isn't among the options provided. Geometry can be tricky, but with the right formulas and a bit of algebra, we can solve these puzzles! Keep practicing, and you'll become a polygon pro in no time!