Polygon Sides: Diagonals Increase By 85 Solved!

Hey math enthusiasts! Ever found yourself pondering the fascinating relationship between the sides and diagonals of polygons? It's a classic geometry problem that can seem tricky at first glance, but fear not! We're going to break it down step by step, making it super easy to understand. In this comprehensive guide, we'll dive deep into the formula for diagonals, explore how changes in the number of sides affect the number of diagonals, and, most importantly, solve the puzzle of how to calculate the sides of a polygon when the diagonals increase by 85. So, grab your thinking caps, and let's get started!

Understanding Polygon Diagonals

Alright, guys, before we jump into the problem, let's make sure we're all on the same page about what diagonals are and how they're calculated. A diagonal is simply a line segment that connects two non-adjacent vertices (corners) of a polygon. Think of a square – it has two diagonals that cross each other in the middle. Now, imagine a pentagon (a five-sided shape); it has even more diagonals! As the number of sides increases, the number of diagonals increases much faster. This is where the formula comes in handy.

The Diagonal Formula

The key to unlocking this puzzle is the formula for calculating the number of diagonals in a polygon. This formula allows us to directly relate the number of sides to the number of diagonals, which is crucial for solving our problem. The formula is:

D = n(n - 3) / 2

Where:

• D represents the number of diagonals.
• n represents the number of sides of the polygon.

This formula might seem a bit cryptic at first, but let's break it down. The term n(n - 3) represents the number of ways to connect each vertex to every other non-adjacent vertex. We subtract 3 from n because we don't want to count the sides of the polygon or the vertex itself. Then, we divide by 2 because each diagonal is counted twice (once for each endpoint).

For example, let's say we have a hexagon (a six-sided shape). Using the formula, we get:

• D = 6(6 - 3) / 2
• D = 6(3) / 2
• D = 18 / 2
• D = 9

So, a hexagon has 9 diagonals. Pretty cool, huh? Now that we have a solid understanding of the diagonal formula, let's move on to the main problem.

Visualizing Diagonals

To truly grasp the concept of diagonals, it's helpful to visualize them. Imagine a series of polygons, starting with a triangle (which has zero diagonals) and moving up to a quadrilateral, a pentagon, a hexagon, and beyond. As you add more sides, you'll notice the diagonals crisscrossing within the shape, forming intricate patterns. This visualization can make the formula feel less abstract and more tangible. You can even draw these polygons yourself and physically count the diagonals to verify the formula – a great way to reinforce your understanding!

The Challenge: Diagonals Increase by 85

Okay, let's get to the heart of the matter. We have a polygon, and somehow, the number of its diagonals has increased by 85. The question is, how do we figure out the original number of sides? This is where our knowledge of the diagonal formula and a bit of algebraic thinking come into play. The core challenge lies in translating the problem's condition—the increase in diagonals—into a mathematical equation that we can solve. This involves not only understanding the formula for diagonals but also recognizing how a change in the number of sides affects the total count of diagonals. We need to consider two scenarios: the polygon's state before the increase in diagonals and its state after the increase. By setting up an equation that reflects this change, we can then use algebraic techniques to solve for the unknown number of sides.

Setting Up the Equation

This is the crucial step in solving the problem. Let's say the original polygon has n sides, and the new polygon has n + x sides (where x is the increase in the number of sides). We know that the number of diagonals in the new polygon is 85 more than the number of diagonals in the original polygon. We can translate this information into an equation using the diagonal formula. We'll have two instances of the diagonal formula: one representing the original polygon and one representing the new polygon. The difference between these two instances will be equal to 85. This equation will be the key to unlocking the solution.

The number of diagonals in the original polygon is:

D₁ = n(n - 3) / 2

Let's assume the number of sides increases to n₂, then the number of diagonals in the new polygon is:

D₂ = n₂(n₂ - 3) / 2

We know that the number of diagonals increased by 85, so:

D₂ = D₁ + 85

However, we don't know n₂ directly. What we do know is that the increase in diagonals is due to an increase in the number of sides. This is a crucial point because it allows us to relate n₂ back to our original unknown, n. Without establishing this relationship, we would have two unknowns in our equation, making it much harder to solve. The problem statement doesn't explicitly tell us how many sides were added, so we need to figure out how many sides must be added to the original polygon for the number of diagonals to increase by 85. This is the missing piece of the puzzle that will allow us to express n₂ in terms of n and create a solvable equation.

Substituting the formulas for D₁ and D₂ and the relationship D₂ = D₁ + 85, we get:

n₂(n₂ - 3) / 2 = n(n - 3) / 2 + 85

This equation looks a bit intimidating, but don't worry! We'll simplify it step by step. The next step is to find the correlation between n and n₂, and that's the trickiest part of the problem. We need to express n₂ in terms of n and a certain value (x), which represents the increase in the sides of the polygon.

Finding the Relationship Between n and n₂

To find the relationship between n and n₂, we need to consider what happens when we increase the number of sides of a polygon. Each new side adds more vertices, and each new vertex can potentially connect to other non-adjacent vertices, creating more diagonals. The key is to understand that the increase in diagonals is not linear with the increase in sides. This means that adding one side doesn't simply add a fixed number of diagonals; the number of diagonals added depends on the existing number of sides. This non-linear relationship is what makes the problem challenging and interesting.

Let's say we increase the sides by x, so:

n₂ = n + x

Now we can substitute n₂ in our equation:

(n + x)(n + x - 3) / 2 = n(n - 3) / 2 + 85

This equation still looks complex, but we're making progress! We've reduced the number of unknowns to just n and x. The next step is to simplify the equation and see if we can solve for n and x. Remember, we're looking for integer solutions because the number of sides of a polygon must be a whole number. This constraint will help us narrow down the possibilities and find the correct answer. The goal now is to manipulate the equation algebraically, expanding the terms, combining like terms, and isolating the variables. This process will reveal the underlying structure of the equation and bring us closer to a solution.

Solving the Equation

Now comes the fun part – the algebra! We need to simplify the equation we derived earlier and solve for n. This will involve expanding the terms, combining like terms, and rearranging the equation into a more manageable form. Algebraic manipulation is a powerful tool in mathematics, allowing us to transform complex expressions into simpler, equivalent forms. In this case, it will help us isolate the variable n and determine its value.

Step-by-Step Simplification

Let's start by multiplying both sides of the equation by 2 to get rid of the fractions:

(n + x)(n + x - 3) = n(n - 3) + 170

Next, we expand the products on both sides:

n² + nx - 3n + nx + x² - 3x = n² - 3n + 170

Now, we can simplify by combining like terms:

n² + 2nx - 3n + x² - 3x = n² - 3n + 170

Notice that n² and -3n appear on both sides of the equation, so we can cancel them out:

2nx + x² - 3x = 170

This is a much simpler equation! Now, we need to think strategically about how to solve for n. We have two unknowns (n and x), but we also have the constraint that both must be integers. This is a key piece of information that we can use to our advantage.

Isolating n

Let's rearrange the equation to isolate n:

2nx = 170 - x² + 3x

Now, divide both sides by 2x:

n = (170 - x² + 3x) / 2x

This equation gives us a direct relationship between n and x. Since n must be an integer, the expression on the right-hand side must also be an integer. This means that (170 - x² + 3x) must be divisible by 2x. This condition significantly narrows down the possible values of x. We can now test different integer values of x to see which ones result in an integer value for n. This is where trial and error, combined with a bit of number sense, comes into play.

Trial and Error with Integer Values

We can start by trying small integer values for x and see if we get an integer value for n. Remember, x represents the increase in the number of sides, so it must be a positive integer. Also, n must be greater than 3, because a polygon needs at least three sides.

Let's try x = 1:

• n = (170 - 1² + 3(1)) / 2(1)
• n = (170 - 1 + 3) / 2
• n = 172 / 2
• n = 86

This gives us an integer value for n, so x = 1 and n = 86 is a potential solution. Let's check if this solution works by calculating the number of diagonals in the original polygon and the new polygon.

Verifying the Solution

If n = 86, the number of diagonals in the original polygon is:

• D₁ = 86(86 - 3) / 2
• D₁ = 86(83) / 2
• D₁ = 3569

If n₂ = n + x = 86 + 1 = 87, the number of diagonals in the new polygon is:

• D₂ = 87(87 - 3) / 2
• D₂ = 87(84) / 2
• D₂ = 3654

The difference in the number of diagonals is:

• D₂ - D₁ = 3654 - 3569 = 85

This confirms that our solution is correct! The original polygon had 86 sides, and by adding 1 side, the number of diagonals increased by 85.

The Answer

So, guys, after all that algebraic maneuvering and careful calculation, we've arrived at the answer! The original polygon had 86 sides. It's pretty amazing how we can use a simple formula and some logical reasoning to solve such an intriguing problem. Geometry and algebra, when combined, offer a powerful toolkit for understanding the world around us.

Final Thoughts

This problem highlights the beauty and interconnectedness of mathematics. We started with a geometric concept – the diagonals of a polygon – and used an algebraic formula to solve a specific problem. Along the way, we employed various problem-solving strategies, including setting up equations, simplifying expressions, and using trial and error. Remember, the key to mastering math is not just memorizing formulas but also understanding the underlying concepts and developing the ability to apply them in different situations.

I hope this explanation has been clear and helpful. If you have any questions or want to explore more challenging polygon problems, feel free to ask! Keep exploring the world of math, and you'll be amazed at what you can discover.