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Chapter 9: Problem 69

In Exercises 69-72, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Pentagon

Short Answer

Expert verified
The number of diagonals in a pentagon is 5.

Step by step solution

01

Identify the Number of Sides

A Pentagon has 5 sides. Hence, in our formula \( n(n-3)/2 \), \( n \) will be 5.
02

Substitute the Value in the Formula

Now, substituting \( n=5 \) into the formula, the expression becomes \[ 5(5-3)/2 \].
03

Calculate the Number of Diagonals

Calculating it further, \[ 5(5-3)/2 \] becomes \[ 5(2)/2 \] which is equal to 5. Hence, a pentagon has 5 diagonals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pentagon
A pentagon is a five-sided polygon, a closed figure with five straight sides and five corners or vertices. It is one of the simplest polygons because it has the fewest number of sides needed to form a closed shape beyond a triangle and a quadrilateral.

Pentagons are very common in both mathematics and the world around us. For example:
  • The famous building known as "The Pentagon" in the United States is shaped like a regular pentagon.
  • Natura艂 occurrences, like sea stars, also exhibit pentagon-like structures.

Understanding pentagons is crucial in geometry. They are the simplest form of polygon that shows us how sides and diagonals work. Becoming familiar with the pentagon is a great starting point for exploring more complex polygon shapes.
Number of Sides of a Polygon
Generally, a polygon is defined by the number of its sides. This characteristic helps determine not only the type of polygon but also properties like the number of diagonals that polygon will have.

Here is a quick guide to understanding common polygons based on their number of sides:
  • Triangles have 3 sides.
  • Quadrilaterals have 4 sides.
  • Pentagons have 5 sides.
  • Hexagons have 6 sides.
  • Heptagons have 7 sides.

The number of sides of a polygon, denoted as \( n \), is also an important variable in formulas related to polygon properties, like calculating the number of diagonals. As the number of sides increases, the complexity and count of possible diagonals increase as well. For example, a pentagon with 5 sides can form 5 diagonals.
Formula for Diagonals in a Polygon
The formula to find the number of diagonals in a polygon is essential for understanding polygon geometry. The formula, \( \frac{n(n-3)}{2}\), is straightforward but powerful. Here, \( n \) represents the number of sides of the polygon.

To apply this formula, follow these steps:
  • Determine the number of sides \( n \) of your polygon.
  • Substitute \( n \) into the formula.
  • Solve the expression to find how many diagonals exist.

For instance, applying this to a pentagon, where \( n = 5 \), substitutes into the formula to get \( \frac{5(5-3)}{2} \) which simplifies to 5, meaning a pentagon has 5 diagonals.

The power of this formula lies in its general applicability. It can be used for any polygon, whether it has 5, 10, or 100 sides, allowing us to easily calculate the number of diagonals just by knowing the number of sides.

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Most popular questions from this chapter

Consider a group of \(n\) people. (a) Explain why the following pattern gives the probabilities that the \(n\) people have distinct birthdays. $$\begin{array}{l}{n=2 : \frac{365}{365} \cdot \frac{364}{365}=\frac{365 \cdot 364}{365^{2}}} \\ {n=3 : \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365}=\frac{365 \cdot 364 \cdot 363}{365^{3}}}\end{array}$$ (b) Use the pattern in part (a) to write an expression for the probability that \(n=4\) people have distinct birthdays. (c) Let \(P_{n}\) be the probability that the \(n\) people have distinct birthdays. Verify that this probability can be obtained recursively by $$P_{1}=1\( and \)P_{n}=\frac{365-(n-1)}{365} P_{n-1}$$ (d) Explain why \(Q_{n}=1-P_{n}\) gives the probability that at least two people in a group of \(n\) people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than \(\frac{1}{2} ?\) Explain.

You are given the probability that an event will happen. Find the probability the event will not happen. \(P(E)=0.87\)

Linear Model, Quadratic Model, or Neither? In Exercises \(61-68\) , write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. $$a_{1}=2$$ $$a_{n}=n-a_{n-1}$$

True or False? In Exercises 77 and 78 , determine whether the statement is true or false. Justify your answer. A sequence with \(n\) terms has \(n-1\) second differences.

You are given the probability that an event will happen. Find the probability the event will not happen. \(P(E)=\frac{1}{4}\)
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