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Chapter 1: Problem 36

Problem 36 A polygon \(A B C D E \cdots\) consists of \(n\) vertices \(A, B, C, D,\) \(E, \ldots,\) and \(n\) sides \(A B, B C, C D, D E \ldots\) which are disjoint except that successive pairs meet at their common endpoint (as when \(A B, B C\) meet at \(B\) ). A polygon is regular if any two sides are congruent (or equal), and any two angles are congruent (or equal). Can a regular \(n\) -gon \(A B C D E \cdots\) always be inscribed in a circle? In other words, does a regular polygon automatically have a "centre", which is equidistant from all \(n\) vertices? \(\triangle\)

Short Answer

Expert verified
Yes, a regular n-gon can always be inscribed in a circle as it has a circumcenter equidistant from all vertices.

Step by step solution

01

Understanding Regular Polygons

A regular polygon is a polygon with all sides and angles equal. This means if we have any regular n-gon, all its sides are the same length, and all its interior angles are equal.
02

Considering a Circle's Properties

A circle is a shape where all points on the surface are equidistant from a center point. If a regular polygon can be inscribed in it, each vertex must lie on the circle's edge, equidistant from the center.
03

Establishing the Circumcenter

For a regular polygon to be inscribed in a circle, it must have a circumcenter, where each vertex of the polygon is equidistant from this point. Regular polygons have symmetrical properties, ensuring that every vertex is equally spaced around the circumcenter.
04

Verifying the Conditions

The symmetry and equal sides/angles of regular polygons ensure they can have a circumcenter. By definition, this circumcenter is the circle's center through which all vertices are equidistantly inscribed.

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Key Concepts

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

Inscribed Circle
An inscribed circle is one that fits perfectly within a polygon, touching each side exactly once. For regular polygons, an inscribed circle can always be drawn, making them uniquely symmetrical. The inscribed circle's center is the same as the polygon's center, around which the polygon's vertices are equidistant.
  • A regular polygon can be inscribed within a circle, meaning all its vertices touch the circle's circumference.
  • The inscribed circle is tangential to each side of the polygon, maintaining equal distance from the polygon's center to the edge.

This property of regular polygons allows them to fit neatly into a circle, emphasizing their uniformity and symmetry.
Circumcenter
The circumcenter of a polygon is a significant point where the perpendicular bisectors of its sides intersect. In regular polygons, this circumcenter is also the center of the circle in which the polygon can be inscribed. This ensures that all vertices of the polygon lie exactly on this circle.
  • In regular polygons, each side's perpendicular bisector connects at one central point, the circumcenter.
  • This circumcenter helps determine the radius of the circle the polygon can be inscribed in.

When all sides and angles of a polygon are equal, the circumcenter is guaranteed to exist and is equidistant from all vertices, offering symmetry and balance.
Polygon Properties
Regular polygons have unique properties that distinguish them from non-regular ones. They have equal sides and equal angles, which leads to specific characteristics like symmetry and the ability to be inscribed in a circle.
  • All sides being equal ensures that no side outpaces its partner, maintaining equilibrium.
  • The equal angles allow regular polygons to have perfect rotational symmetry.
Such properties make regular polygons predictable and geometric patterns stable, allowing easy tiling or tesselling to form consistent designs.
Geometric Symmetry
Geometric symmetry in polygons refers to their balanced proportions where each side mirrors its opposite counterpart. In regular polygons, this symmetry means each rotation by an angle equivalent to one of the internal angles is indistinguishable from its original position.
  • Regular polygons have both rotational and reflectional symmetry.
  • Every axis passing through the center exactly halves the regular polygon, highlighting symmetrical balance.
This symmetry not only characterizes regular polygons but also contributes to their aesthetic appeal and functionality in various design and architectural applications.

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Most popular questions from this chapter

Problem \(29(A+B=C)\) The 3 by 1 rectangle \(A D E H\) consists of three adjacent unit squares: \(A B G H, B C F G, C D E F\) left to right, with \(A\) in the top left corner. Prove that $$ \angle D A E+\angle D B E=\angle D C E $$

(a) Expand and simplify in your head: (i) \((\sqrt{2}+1)^{2}\) (ii) \((\sqrt{2}-1)^{2}\) (iii) \((1+\sqrt{2})^{3}\) (b) Simplify: (i) \(\sqrt{10+4 \sqrt{6}}\) (ii) \(\sqrt{5+2 \sqrt{6}}\) (iii) \(\sqrt{\frac{3+\sqrt{5}}{2}}\) (iv) \(\sqrt{10-2 \sqrt{5}}\) The expressions which occur in exercises to develop fluency in working with surds often appear arbitrary. But they may not be. The arithmetic of surds arises naturally: for example, some of the expressions in the previous problem have already featured in Problem \(\mathbf{3}(\mathrm{c}) .\) In particular, surds will feature whenever Pythagoras' Theorem is used to calculate lengths in geometry, or when a proportion arising from similar triangles requires us to solve a quadratic equation. So surd arithmetic is important. For example: \- A regular octagon with side length 1 can be surrounded by a square of side \(\sqrt{2}+1\) (which is also the diameter of its incircle); so the area of the regular octagon equals \((\sqrt{2}+1)^{2}-1\) (the square minus the four corners). \- \(\sqrt{2}-1\) features repeatedly in the attempt to apply the Euclidean algorithm, or anthyphairesis, to express \(\sqrt{2}\) as a "continued fraction". -\(\sqrt{10-2 \sqrt{5}}\) may look like an arbitrary, uninteresting repeated surd, but is in fact very interesting, and has already featured as \(4 \sin 36^{\circ}\) in Problem \(\mathbf{3}(\mathrm{c})\) \- One of the simplest ruler and compasses constructions for a regular pentagon \(A B C D E\) (see Problem 185) starts with a circle of radius 2 , centre \(O,\) and a point \(A\) on the circle, and in three steps constructs the next point \(B\) on the circle, where \(\underline{A B}\) is an edge of the inscribed regular pentagon, and $$ \underline{A B}=\sqrt{10-2 \sqrt{5}} $$

Which is bigger: \(17 \%\) of nineteen million, or \(19 \%\) of seventeen million?

(a)(i) Explain why any integer that is a factor (or a divisor) of both \(m\) and \(n\) must also be a factor of their difference \(m-n,\) and of their sum \(m+n\). (ii) Prove that $$ H C F(m, n)=H C F(m-n, n) $$ (iii) Use this to calculate in your head \(H C F(1001,91)\) without factorising either number. (b)(i) Prove that: \(H C F(m, m+1)=1\). (ii) Find \(H C F(m, 2 m+1)\). (iii) Find \(H C F\left(m^{2}+1, m-1\right)\).

(a) Evaluate $$ \left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right) . $$ (b) Evaluate $$ \sqrt{1+\frac{1}{2}} \times \sqrt{1+\frac{1}{3}} \times \sqrt{1+\frac{1}{4}} \times \sqrt{1+\frac{1}{5}} \times \sqrt{1+\frac{1}{6}} \times \sqrt{1+\frac{1}{7}} $$ (c) We write the product " \(4 \times 3 \times 2 \times 1 "\) as "4!" (and we read this as "4 factorial"). Using only pencil and paper, how quickly can you work out the number of weeks in \(10 !\) seconds? \(\Delta\)
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