Determine The Number Of Sides Of A Convex Polygon Where The Difference Between The Total Number Of Segments Formed By Two Non-consecutive Points And The Total Number Of Segments Formed By Two Consecutive Points Is 25.
Introduction
In geometry, a polygon is a two-dimensional shape formed by a sequence of straight line segments called sides or edges. Polygons are classified based on the number of sides they have, such as triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and so on. A convex polygon is a polygon in which no interior angle is greater than 180 degrees, meaning that all its interior angles point outwards. Understanding the properties of polygons, especially convex polygons, is crucial in various fields, including architecture, engineering, and computer graphics.
This article addresses a specific problem related to convex polygons: determining the number of sides of a convex polygon based on the difference between the total number of segments formed by non-consecutive points and the total number of segments formed by consecutive points. This involves understanding the formulas for calculating the total number of diagonals and sides in a polygon. The problem combines basic geometric principles with algebraic manipulation, providing a comprehensive exercise in mathematical reasoning.
Understanding Polygons and Their Properties
To solve the problem, it's essential to first grasp the fundamental properties of polygons. A polygon is a closed shape made up of straight line segments. The points where these segments meet are called vertices. The line segments connecting the vertices are the sides of the polygon. In a convex polygon, all interior angles are less than 180 degrees, meaning that any line segment connecting two points inside the polygon lies entirely within the polygon. This property simplifies many calculations related to diagonals and interior angles.
One key concept in understanding polygons is the diagonal. A diagonal is a line segment that connects two non-adjacent vertices of the polygon. The number of diagonals in a polygon is a crucial characteristic that helps distinguish different types of polygons. The formula to calculate the number of diagonals (D) in a polygon with n sides is given by:
D = n(n - 3) / 2
This formula arises from the fact that each vertex can connect to every other vertex except itself and its two adjacent vertices. However, since each diagonal connects two vertices, we divide by 2 to avoid double counting. Understanding this formula is essential for solving problems related to the number of diagonals in a polygon.
Consecutive and Non-Consecutive Points
In the context of this problem, it is important to differentiate between segments formed by consecutive points and those formed by non-consecutive points. Consecutive points are vertices that are directly adjacent to each other, forming the sides of the polygon. For instance, in a pentagon (5-sided polygon), if we label the vertices A, B, C, D, and E, then AB, BC, CD, DE, and EA are segments formed by consecutive points, which are the sides of the pentagon.
On the other hand, non-consecutive points are vertices that are not directly adjacent to each other. Segments formed by non-consecutive points are the diagonals of the polygon. Using the same pentagon example, segments AC, AD, BD, BE, and CE are formed by non-consecutive points. The problem focuses on the difference between the total number of these two types of segments. This distinction is crucial for setting up the correct equations and solving for the number of sides.
The Problem Statement
The problem asks us to determine the number of sides of a convex polygon where the difference between the total number of segments formed by non-consecutive points (diagonals) and the total number of segments formed by consecutive points (sides) is equal to 25. Mathematically, this can be expressed as:
Number of Diagonals - Number of Sides = 25
Given that the number of sides is n and the number of diagonals is n(n - 3) / 2, the equation can be written as:
n(n - 3) / 2 - n = 25
Solving this equation for n will give us the number of sides of the polygon. The solution involves algebraic manipulation, including simplifying the equation, rearranging terms, and solving the resulting quadratic equation. This problem effectively tests one's understanding of geometric properties and algebraic skills.
Setting Up the Equation
To solve the problem, the initial step involves setting up the equation that represents the given condition. The problem states that the difference between the total number of segments formed by non-consecutive points (diagonals) and the total number of segments formed by consecutive points (sides) is equal to 25. Let's denote the number of sides of the convex polygon as n. As discussed earlier, the number of segments formed by consecutive points is simply the number of sides, which is n.
Expressing the Number of Diagonals
The number of segments formed by non-consecutive points is the number of diagonals. The formula for the number of diagonals (D) in a polygon with n sides is given by:
D = n(n - 3) / 2
This formula is derived from the fact that each vertex can connect to every other vertex except itself and its two adjacent vertices. Dividing by 2 accounts for the fact that each diagonal connects two vertices, preventing double counting. Understanding this formula is critical for setting up the correct equation.
Formulating the Equation
The problem states that the difference between the number of diagonals and the number of sides is 25. Therefore, we can write the equation as:
Number of Diagonals - Number of Sides = 25
Substituting the expressions for the number of diagonals and the number of sides, we get:
[n(n - 3) / 2] - n = 25
This equation represents the relationship between the number of sides n and the given condition. The next step involves solving this equation for n to find the number of sides of the polygon. This requires algebraic manipulation to simplify the equation and solve for the unknown variable.
Simplifying the Equation
Before solving for n, it's essential to simplify the equation to make it easier to work with. The equation is:
[n(n - 3) / 2] - n = 25
To eliminate the fraction, we can multiply the entire equation by 2:
2 * [[n(n - 3) / 2] - n] = 2 * 25
This simplifies to:
n(n - 3) - 2n = 50
Next, we expand the term n(n - 3):
n^2 - 3n - 2n = 50
Combining like terms, we get:
n^2 - 5n = 50
To solve for n, we need to rearrange the equation into the standard quadratic form, which is ax^2 + bx + c = 0. Subtracting 50 from both sides, we get:
n^2 - 5n - 50 = 0
This is now a quadratic equation in the standard form. The next step is to solve this quadratic equation for n.
Solving the Quadratic Equation
After setting up and simplifying the equation, we arrived at the quadratic equation:
n^2 - 5n - 50 = 0
To solve this quadratic equation for n, we can use several methods, such as factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach.
Factoring the Quadratic Equation
We need to find two numbers that multiply to -50 and add to -5. These numbers are -10 and 5. Therefore, we can factor the quadratic equation as:
(n - 10)(n + 5) = 0
This factored form allows us to find the possible values for n by setting each factor equal to zero:
n - 10 = 0 or n + 5 = 0
Solving these two equations gives us:
n = 10 or n = -5
Interpreting the Solutions
We have two possible solutions for n: 10 and -5. However, since n represents the number of sides of a polygon, it must be a positive integer. Therefore, the solution n = -5 is not valid in this context. The only valid solution is n = 10.
The Number of Sides
Based on the solution of the quadratic equation, we conclude that the number of sides of the convex polygon is 10. This means the polygon is a decagon. To verify our solution, we can substitute n = 10 back into the original equation and check if it holds true.
Verification and Conclusion
After solving the quadratic equation, we found that the number of sides of the convex polygon, n, is 10. To ensure the correctness of our solution, we need to verify it by substituting n = 10 back into the original equation.
Verifying the Solution
The original equation was:
[n(n - 3) / 2] - n = 25
Substituting n = 10, we get:
[10(10 - 3) / 2] - 10 = 25
Simplifying the expression:
[10(7) / 2] - 10 = 25
[70 / 2] - 10 = 25
35 - 10 = 25
25 = 25
Since the equation holds true, our solution n = 10 is correct. This confirms that the convex polygon has 10 sides, making it a decagon.
Conclusion
In conclusion, we determined the number of sides of a convex polygon where the difference between the total number of segments formed by non-consecutive points and the total number of segments formed by consecutive points is equal to 25. By setting up and solving the equation [n(n - 3) / 2] - n = 25, we found that the number of sides n is 10. This problem demonstrates the application of geometric principles and algebraic techniques to solve problems involving polygons. Understanding the properties of polygons, such as the number of diagonals and the relationship between sides and vertices, is crucial for solving such problems. Additionally, the ability to manipulate algebraic equations and solve quadratic equations is essential for arriving at the correct solution. The final answer, 10 sides, signifies that the polygon in question is a decagon.