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Chapter 5: Problem 4

a) Are any two equilateral hexagons similar? b) Are any two regular hexagons similar?

Short Answer

Expert verified
a) No, equilateral hexagons are not always similar. b) Yes, regular hexagons are always similar.

Step by step solution

01

Understanding Equilateral Hexagons

Equilateral hexagons have all sides equal but not necessarily all angles equal. Therefore, they may not be similar as similarity requires both angles and proportional sides.
02

Analyzing Similarity of Equilateral Hexagons

Two polygons are similar if corresponding angles are equal and corresponding sides are proportional. Equilateral hexagons can have different angles without maintaining proportional sides, so they are not always similar.
03

Understanding Regular Hexagons

Regular hexagons have all sides equal and all angles equal, with each internal angle being 120 degrees. This regularity is a key aspect of their geometry.
04

Analyzing Similarity of Regular Hexagons

Since all regular hexagons have equal angles and sides in a fixed ratio, any two regular hexagons are always similar regardless of their sizes.

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Key Concepts

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

Equilateral Hexagons
Equilateral hexagons are interesting shapes in geometry. Each has six sides of equal length, but unlike regular hexagons, their angles can differ. This variability in their angles means that equilateral hexagons can take on numerous shapes.
  • All six sides are of equal length.
  • The internal angles can vary.
For two polygons to be similar, they must have equal corresponding angles and proportional corresponding sides. Equilateral hexagons can change their angle sizes without changing the side lengths, which means they do not necessarily have equal angles.
Therefore, not all equilateral hexagons are similar.
Regular Hexagons
Regular hexagons are a special kind of hexagon where both sides and angles are uniformly equal. This uniformity simplifies many of their properties and interactions with other shapes. Regular hexagons have a fixed internal angle of 120 degrees each.
  • All sides are equal.
  • Each internal angle measures 120 degrees.
This property makes regular hexagons very predictable in terms of geometry. The combination of equal sides and angles ensures that any two regular hexagons, regardless of their size, are always similar.
Polygon Similarity
Polygon similarity is a key aspect in understanding relationships between different polygons. Two polygons are similar if their corresponding angles are equal and the ratios of their corresponding side lengths are proportional.
  • Corresponding angles must be the same for similarity.
  • Sides must be in proportional lengths.
In the case of regular hexagons, they automatically satisfy these conditions because all of their angles and side ratios are consistent. On the other hand, equilateral hexagons do not always meet these criteria because while their sides are equal, their angles can differ. This difference in angles makes equilateral hexagons not similar by default.

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