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Chapter 11: Problem 34

Tell whether the statement is true or false. Explain your reasoning The apothem of a regular polygon is always less than the radius.

Short Answer

Expert verified
The statement is True. The apothem of a regular polygon is always less than the radius, since the radius extends to a vertex, while the apothem ends at the midpoint of a side.

Step by step solution

01

Identify the Concepts

The radius of a regular polygon is defined as the distance from the center of the polygon to one of its vertices. The apothem, on the other hand, is the distance from the center of the polygon to the midpoint of any side.
02

Visualize the Polygon

Understanding the definitions well, visualize a regular polygon. Draw the radius and the apothem in it. It is observed that the apothem ends halfway along a side of the polygon, while the radius extends to a vertex, thus the radius is longer.
03

Understand the Relationship between the Radius and the Apothem

In a regular polygon, the apothem always forms a right angle with the side of the polygon where it ends. Therefore, it becomes the side of a right triangle where the other side is half the length of a side of the polygon, and the hypotenuse is the radius of the polygon. According to Pythagoras theorem, the square of the hypotenuse (radius) is equal to the sum of the squares of the other two sides. Hence the hypotenuse (radius) is always longer than any other side (apothem).
04

Verdict on the statement

Based on the breakdown, the statement 'The apothem of a regular polygon is always less than the radius' is True. The radius always extends to a vertex, while the apothem ends halfway along a side of the polygon, thus the radius is always longer than the apothem.

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Key Concepts

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

Understanding the Apothem
The apothem of a regular polygon is a crucial geometric element that helps in calculating the area and other properties of the polygon. It is defined as the line segment from the center of the polygon to the midpoint of one of its sides. The apothem is always perpendicular to the side it meets, forming a right angle.

  • It serves as one of the three sides of a right triangle, where one side lies along the polygon's side, meeting the apothem at its midpoint.
  • The apothem provides a key insight into the inner structure of the polygon, revealing symmetrical properties.
  • It is especially important in regular polygons where it can be used in area calculations since the area of a regular polygon can be expressed as \( \frac{1}{2} \times \text{perimeter} \times \text{apothem} \).
Understanding the concept of apothem deepens the grasp of geometric principles, making it easier to handle complex problems involving polygons.
Radius of a Polygon
The radius of a regular polygon plays a pivotal role in defining its geometry. It is the line segment that extends from the center of the polygon to one of its vertices. This segment is essential in understanding the polygon's overall shape and size.

  • The radius is longer than the apothem because it reaches a vertex rather than stopping halfway along a side.
  • It forms the hypotenuse of a right triangle in configurations that include the apothem and half a side of the polygon, following the Pythagorean theorem.
  • This measurement is crucial for calculating the circumcircle, the circle that passes through all the vertices of the polygon.
Through the radius, one can grasp the external expanse of the polygon, providing a sense of how the shape spreads outward from its center.
Right Triangle in Polygons
In many cases, particularly when dealing with regular polygons, the concept of a right triangle can be a powerful tool. When you draw a radius and an apothem inside a regular polygon, you form a right triangle.

  • One side of the right triangle is the apothem, which is perpendicular to one of the polygon's sides.
  • The radius acts as the hypotenuse, stretching from the center to a vertex.
  • The third side is half of the polygon's side between two adjacent vertices.
This configuration allows for the use of the Pythagorean theorem, which states: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse (radius). Using this relationship simplifies calculating unknown lengths and validates geometric principles regarding shapes.

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Most popular questions from this chapter

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