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Chapter 9: Problem 9

For each exercise draw a circle and inscribe the polygon in the circle. A parallelogram

Short Answer

Expert verified
The parallelogram can be inscribed in a circle by choosing the diameter of the circle as the base of the parallelogram and then effectively choosing the other two vertices on the circle so that they are equidistant from the base's endpoints.

Step by step solution

01

Draw a Circle

To start, draw a circle. The size is not important as long as there is enough space inside for a parallelogram to be drawn.
02

Draw a Diameter of the Circle

Next, draw a straight line through the center of the circle. This line is called a diameter of the circle and will serve as the base for the parallelogram.
03

Determine the Vertices of the Parallelogram

Accept the endpoints of the diameter as the first two vertices of the parallelogram. To get the other two vertices, set two points on the circle so that they are equidistant from the endpoints of the diameter.
04

Draw the Parallelogram

Draw lines to connect the vertices. Make sure that opposite sides are parallel to each other, because that's a requirement for a figure to be a parallelogram.
05

Verify

Ensure that the parallelogram adequately is within the circle. All the vertices of the parallelogram must touch the circle.

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Key Concepts

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

Inscribed Polygons
When you hear the term "inscribed polygon," it essentially means a polygon that is drawn inside a circle in such a way that all its vertices (corners) touch the circle. This circle is often called the circumcircle.
Imagine a triangle, a square, or any other polygon sketched neatly inside a circle with each corner of the polygon just brushing the edge. That's what inscribed means!
  • Vertices of the polygon lie on the circumference of the circle.
  • Such polygons can be triangles, quadrilaterals, pentagons, etc.
  • An inscribed polygon is sometimes referred to as a cyclic polygon.
Understanding inscribed polygons is fundamental in the study of geometry as they help us explore the relationships between angles and sides in more complex shapes. Furthermore, they set the ground for exploring several geometric theorems and properties related to circles.
Parallelogram
The parallelogram is a simple yet intriguing figure in geometry. Defined as a four-sided shape where opposite sides are parallel and equal in length, it possesses a few special properties. Think of a simple quadrilateral with its opposite sides always maintaining that uniform distance parallel to each other.
  • Opposite sides are equal in length.
  • Opposite angles are equal.
  • The diagonals of a parallelogram bisect each other.
  • The sum of the angles in a parallelogram is always 360 degrees.
These properties make parallelogram a versatile shape often used in various mathematical problems. In this particular exercise, the challenge lies in inscribing a parallelogram inside a circle, which can lead to different types of parallelograms such as rhombuses or rectangles.
Circle Properties
A circle, with its perfect roundness, is a fascinating figure in geometry, coming with its unique set of properties. The most prominent feature of a circle is its center point, from which every point on the circle is equidistant. This distance is known as the radius.
  • The diameter is twice the radius and it passes through the center.
  • A chord is any line segment with both endpoints on the circle.
  • The circumference represents the distance around the circle.
  • The area of a circle can be calculated as \( \pi r^2 \).
In the context of inscribing a polygon, such as a parallelogram, understanding these properties can be crucial. They help determine how shapes can harmoniously lie within the circle, ensuring that their vertices touch the boundary seamlessly. The ability to understand and utilize circle properties provides a strong foundation for tackling more complex geometric challenges.

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Most popular questions from this chapter

The latitude of a city is given. Sketch the Earth and a circle of latitude through the city. Find the radius of this circle. Columbus, Ohio; \(40^{\circ} \mathrm{N}\)

Sketch a circle \(O\) with radius 10 and chord \(\overline{X Y} 8 \mathrm{cm}\) long. How far is the chord from \(O ?\)

Use trigonometry to find the measure of the arc cut off by a chord \(12 \mathrm{cm}\) long in a circle of radius \(10 \mathrm{cm} .\)

The latitude of a city is given. Sketch the Earth and a circle of latitude through the city. Find the radius of this circle. Sydney, Australia; \(34^{\circ} \mathrm{S}\)

At 11 o'clock the hands of a clock form an angle of \(\underline{?}\circ\) .
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