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Chapter 7: Problem 46

THOUGHT PROVOKING Create a regular hexagon using congruent parallelograms.

Short Answer

Expert verified
A regular hexagon can be generated by creating six congruent parallelograms. Draw one parallelogram and then connect another five in a way that each shares a side with the last, forming a full circle. The result will be a regular hexagon.

Step by step solution

01

Constructing the initial parallelogram

Start by drawing a line segment that will serve as the base for your regular hexagon. Then draw another line, of the same length, aligned at an angle of 30 degrees to one side of your base. Attach another line from the end point of the second line to the end of the base. This will create the first parallelogram.
02

Repeating the parallelograms

Next, replicate this parallelogram by drawing a line of the same length as the base of the first parallelogram, attached to the end of the base line of the first parallelogram but angled at 30 degrees in the opposite direction. Connect the unjoined vertices to form the second parallelogram.
03

Finishing the hexagon

Repeat the process of creating congruent parallelograms until six are attached at their sides forming a closed figure. The resultant shape is a regular hexagon.

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Key Concepts

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

Regular Hexagon
A regular hexagon is a six-sided polygon where all sides and angles are equal. Each internal angle in a regular hexagon is 120 degrees. Hexagons are a common shape in nature, such as in honeycombs. When creating a regular hexagon using congruent parallelograms, each parallelogram contributes to the symmetry and uniformity of the hexagon.
The key characteristic of a regular hexagon is its rotational symmetry. Rotate it by 60 degrees around its center, and it looks the same.
  • The sides of the hexagon are all of equal length.
  • All internal angles are 120 degrees.
  • It is symmetric and can be divided into 6 equilateral triangles.
Understanding the regular hexagon helps in many applications, such as tiling and constructing complex structures.
Parallelogram
A parallelogram is a simple quadrilateral with opposite sides that are equal and parallel. This quality ensures that opposite angles are equal too. Parallelograms are essential in constructing complex shapes like hexagons.
Some important properties to remember are:
  • Opposite sides are parallel and equal in length.
  • Opposite angles are equal.
  • The diagonals bisect each other.
In the context of creating a hexagon, you use parallelograms' repetitive and symmetrical nature to form each side of the hexagon. This method exploits the equal side lengths and angles to ensure that each piece fits perfectly together.
Congruent Polygons
Congruent polygons are figures that have the same shape and size. In other words, all corresponding sides and interior angles are equal. This concept is critical when creating shapes like hexagons where uniformity is desired for accuracy and aesthetics.
Key characteristics are:
  • Equal corresponding side lengths.
  • Identical corresponding angles.
  • They can be perfectly superimposed onto each other.
Using congruent parallelograms ensures that each section of the hexagon is indistinguishable from the others, maintaining balance and symmetry. By using congruency, one ensures precision in geometric constructions.
Angle Construction
An integral part of forming complex polygons like regular hexagons is angle construction. It's the method of creating specific angles by drawing lines at calculated distances and positions. A regular hexagon has internal angles of 120 degrees, but when constructing one using parallelograms, careful attention must be given to smaller angles, like the 30 degrees between the base of each parallelogram and its adjacent side.
When working with angles:
  • Start with a precise measurement tool like a protractor.
  • Understand that the sum of the internal angles in a hexagon is 720 degrees.
  • Breaking the hexagon into equilateral triangles helps identify correct angle measures for construction.
This precision is essential for ensuring the completed hexagon's sides perfectly align without gaps or overlaps. Angle construction thus provides a solid foundation for accurate geometric designs.

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Most popular questions from this chapter

CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. A square is _____ a rhombus.

CRITICAL THINKING In Exercises \(65-70\) , complete each statement with always, sometimes, or never. Explain your reasoning. A rectangle _____ has perpendicular diagonals.

CRITICAL THINKING Which quadrilateral could be called a regular quadrilateral? Explain your reasoning.

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MAKING AN ARGUMEN POur friend claims that to I nd the interior angle measures of a regular polygon, you do not have to use the Polygon Interior Angles Theorem (Theorem 7.1). You instead can use the Polygon Exterior Angles Theorem (Theorem 7.2) and then the Linear Pair Postulate (Postulate 2.8 ). Is your friend correct? Explain your reasoning.
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