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Chapter 1: Problem 3

Draw an example of each polygon. Octagon

Short Answer

Expert verified
An octagon is a polygon with 8 sides and can be drawn by connecting 8 lines end to end with 135-degree angles between them.

Step by step solution

01

- Understanding an Octagon

An octagon is a polygon with 8 sides and 8 angles. Each side should be of equal length if it's a regular octagon.
02

- Drawing the First Side

Start by drawing a straight line of reasonable length. This will be the first side of the octagon.
03

- Drawing the Subsequent Sides

Draw the second side at a 135-degree angle from the first side. Continue drawing each subsequent side at a 135-degree angle from the previous one until you have eight sides.
04

- Completing the Octagon

After drawing the eight sides, ensure all sides are connected to form a closed shape. Verify that there are exactly 8 sides and 8 angles.

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Key Concepts

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

geometry basics
Geometry is an essential branch of mathematics that deals with shapes, sizes, and the properties of space. Understanding basic geometry helps in identifying different types of polygons and their characteristics.
  • Shapes and Angles: Basic shapes include triangles, squares, and circles, each with different angle properties.
  • Lines and Segments: Lines extend infinitely in both directions, while line segments have defined endpoints.
  • Polygons: These are shapes with multiple straight sides, like triangles (3 sides), quadrilaterals (4 sides), and octagons (8 sides).
Each type of polygon has specific properties that define its shape and structure.
polygon properties
Polygons are multi-sided figures with some unique properties. Understanding these properties makes it simpler to draw and identify polygons.
  • Regular vs Irregular: Regular polygons have equal side lengths and equal angles, while irregular ones do not.
  • Convex vs Concave: Convex polygons have all interior angles less than 180 degrees, making them 'bulge' outward. Concave polygons have at least one interior angle greater than 180 degrees, causing them to 'cave in'.
  • Sum of Interior Angles: The formula to find the sum of interior angles of a polygon is \( (n-2) \times 180^\text{\circ} \), where 'n' is the number of sides. For an octagon, this sum is \( (8-2) \times 180^\text{\circ} = 1080^\text{\circ} \).
Octagons are particularly interesting because they are eight-sided polygons with rich properties that make them common in architecture and design.
step-by-step drawing
Drawing an octagon might seem challenging, but with step-by-step guidance, it becomes manageable. Follow these steps:
  • Step 1 - Understanding an Octagon: Know that an octagon has 8 sides and 8 angles. For a regular octagon, each side is of equal length, and each internal angle measures 135 degrees.
  • Step 2 - Drawing the First Side: Start by drawing a straight line. This will be your first side.
  • Step 3 - Drawing the Subsequent Sides: Draw each subsequent side at a 135-degree angle from the previous side. Repeat this process until you have drawn all 8 sides.
  • Step 4 - Completing the Octagon: Ensure that all 8 sides connect to form a closed shape. Double-check to confirm that your octagon has 8 equal sides and 8 equal angles.
Practice makes perfect, so keep trying until you are satisfied with your octagon!

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Most popular questions from this chapter

Given two points \(A\) and \(B\), there is only one segment that you can name: \(A \bar{B}\). With three collinear points \(A, B,\) and \(C,\) there are three different segments that you can name: \(\overline{A B}, \overline{A C},\) and \(\overline{B C}\). With five collinear points \(A, B, C, D,\) and \(E,\) how many different segments can you name?

If the signs of the coordinates of collinear points \(P(-6,-2), Q(-5,2),\) and \(R(-4,6)\) are reversed, are the three new points still collinear? Draw a picture and explain why. (GRAPH CAN'T COPY).

Adjacent Angles \(\angle X Q A\) and \(\angle X Q Y\) share a vertex and a side. Taken together they form the larger angle \(\angle A Q Y\). Compare their measures. Does \(m \angle X Q A+m \angle X Q Y=m \angle A Q Y ?\)

If points \(A, B,\) and \(C\) are collinear and \(B\) is between \(A\) and \(C,\) then \(A B+B C=A C .\) This is called segment addition. Solve the following problem and explain how it represents segment addition. Podunkville, Smallville, and Gotham City lie along a straight highway with Smallville between the other two towns. If Podunkville and Smallville are \(70 \mathrm{km}\) apart and Smallville and Gotham City are \(110 \mathrm{km}\) apart, how far apart are Podunkville and Gotham City?

Tell whether the statement is true or false. For each false statement, sketch a counterexample or explain why the statement is false. If two lines intersect to form a right angle, then the lines are perpendicular.
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