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Chapter 4: Problem 46

Find the length of the sides of a regular hexagon inscribed in a circle of radius 25 inches.

Short Answer

Expert verified
The length of each side of the hexagon is 25 inches.

Step by step solution

01

Draw a diagram

To visualize the problem, draw a circle with a radius of 25 inches and inscribe a hexagon within it. Connect the center of the circle to the corner points of the hexagon to divide it into six equilateral triangles.
02

Use geometric properties of the hexagon and circle

Since the hexagon is regular and is inscribed in a circle, it can be divided into six equal equilateral triangles, with the center of the circle as their common point. The radius of the circle is also the side of the equilateral triangle.
03

Conclusion

Given that the radius of the circle, which is also one side of each equilateral triangle, measures 25 inches, the length of each side of the hexagon is also 25 inches.

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Key Concepts

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

Equilateral Triangles
When a regular hexagon is inscribed in a circle, it can be beautifully divided into six equilateral triangles. An equilateral triangle is a special kind of triangle where all three sides have the same length and all three angles are equal, precisely 60 degrees each.
In the case of a hexagon inscribed in a circle, each of the triangles formed by connecting the center of the circle to the vertices of the hexagon has the same side length as the radius of the circle. This makes the triangles not just equilateral, but isotropic around the circle, resulting in perfect symmetry.
To summarize:
  • All sides of the equilateral triangles are equal to the radius of the circle.
  • Each interior angle of the equilateral triangle measures 60 degrees.
  • These triangles partition the hexagon into six spaces of equal area.
Inscribed Circle
An inscribed circle is one where all the vertices of a polygon touch the circle. When we talk about a regular hexagon inscribed in a circle, we're referring to a scenario where the circle perfectly fits around the hexagon.
This relationship between the circle and hexagon leads to some interesting geometric properties. A circle that inscribes a regular hexagon will always have a radius that is equal to the length of the sides of the hexagon.
Key points to remember:
  • The radius of the circle is equal to the side length of the hexagon.
  • All six vertices of the hexagon lie on the circle's circumference.
  • Because the hexagon is regular, each side is the same length, reinforcing the radial symmetry with respect to the circle.
Geometric Properties
The properties of a regular hexagon and its relation to circles bring about a range of geometric insights. Begin by recognizing that a regular hexagon has equal-length sides and angles. When inscribed in a circle, this geometry becomes even more unique:
  • Each angle inside the hexagon is 120 degrees because the hexagon spreads out equally around the central point.
  • The circle connecting all vertices acts as a unifying feature, ensuring each triangle is identical.
  • This symmetry makes calculations simpler, as you can leverage the repetitive nature of equilateral triangles and consistent angles.
Understanding these properties helps visualize and solve problems where regular hexagons and circles overlap. Recognize how symmetry plays a role in making these patterns predictable and understandable.

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Most popular questions from this chapter

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow \infty, \text { the value of } \arctan x \rightarrow$$ \(\square\).

Graph \(f\) and \(g\) in the same coordinate plane. Include two full periods. Make a conjecture about the functions. $$f(x)=\sin x, \quad g(x)=\cos \left(x-\frac{\pi}{2}\right)$$

Consider the function \(f(x)=x-\cos x\) (a) Use a graphing utility to graph the function and verify that there exists a zero between 0 and 1 . Use the graph to approximate the zero. (b) Starting with \(x_{0}=1,\) generate a scquence \(x_{1}, x_{2}\) \(x_{3}, \ldots,\) where \(x_{n}=\cos \left(x_{n-1}\right) .\) For example, \(x_{0}=1\) \(x_{1}=\cos \left(x_{0}\right)\) \(x_{2}=\cos \left(x_{1}\right)\) \(x_{3}=\cos \left(x_{2}\right)\) \(\vdots\) What value does the sequence approach?

Find two solutions of each equation. Give your answers in degrees \(\left(0^{\circ} \leq \theta<360^{\circ}\right)\) and in radians \((0 \leq \theta<2 \pi) .\) Do not use a calculator. (a) \(\sin \theta=\frac{\sqrt{3}}{2}\) (b) \(\sin \theta=-\frac{\sqrt{3}}{2}\)

Fill in the blank. If not possible, state the reason. As \(x \rightarrow 1^{-},\) the value of arccos \(x \rightarrow\) \(\square\).
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