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Chapter 9: Problem 6

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

Short Answer

Expert verified
A rectangle can be inscribed in a circle by drawing the circle and diameter, using the diameter as a diagonal of the rectangle, and drawing the sides of the rectangle perpendicular to the diameter so that they touch the circle at exactly four points.

Step by step solution

01

Draw a Circle

Begin by drawing a circle. This will be the boundary inside which the rectangle will be inscribed. The center of the circle will be the point where the diagonals of the rectangle intersect.
02

Draw the Diameter of the Circle

Draw a straight line passing through the center of the circle. This is called the diameter of the circle. This line will also represent the diagonal of the rectangle that we want to inscribe in the circle.
03

Mark the Rectangle Vertices

Using the endpoints of the diameter as the vertices, draw perpendicular lines from the diameter at those points. Remember, in a rectangle all angles are right angles which means the angles here are 90 degrees. These perpendicular lines form the sides of the rectangle.
04

Complete the Rectangle

Complete the rectangle by connecting the open ends of the perpendicular lines. Make sure the lines you draw are touching the circle at exactly two points each because this ensures that the rectangle is inscribed correctly inside the circle.

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Key Concepts

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

Inscribed Polygon
An inscribed polygon is a figure whose vertices lie on a circle's circumference. The circle is referred to as the circumcircle of the polygon. For any given polygon, the circumcircle is the smallest circle that can entirely contain it. To understand this better, imagine placing a polygon inside a circle such that every corner of the polygon touches the circle. This is what it means for a polygon to be inscribed.
  • The circle provides a perfect boundary for the polygon.
  • All the vertices of the polygon touch the circle's boundary.
  • For rectangles, the diagonals must also act as diameters of the circle.
By ensuring the vertices touch the circle, we secure that the polygon is optimally positioned within the circle, maintaining geometric harmony and balance.
Circle
A circle is a simple, closed shape where every point on its boundary is equidistant from a fixed point known as the center. The distance from the center to any point on the circle is known as the radius.
When dealing with inscribed rectangles, the circle's properties serve crucial geometric functions.
  • The diameter of the circle is the longest chord and passes through the center.
  • When a rectangle is inscribed, its diagonal equals the diameter of the circle.
  • This relationship helps in determining whether the rectangle fits perfectly into the circle.
Understanding these foundational properties of a circle is key to effectively leveraging its symmetry and geometry to inscribe other shapes into it.
Rectangle
A rectangle is a four-sided polygon, also known as a quadrilateral, where each angle is a right angle (90 degrees). Typically characterized by opposite sides being equal in length, rectangles display a symmetrical shape that aligns well within circular boundaries like those of an inscribed circle.
  • The diagonals of a rectangle are always equal.
  • When inscribed in a circle, these diagonals become the circle's diameter.
  • This also ensures that the rectangle maintains its property of equal opposite sides.
When you inscribe a rectangle in a circle, you are effectively using the geometric rules to make sure every corner or vertex of the rectangle touches the circle's boundary, creating a perfect geometric and visual balance.
Perpendicular Lines
Perpendicular lines occur when two lines meet at a right angle (90 degrees). This is a fundamental concept in the construction of shapes like rectangles. For instance, when inscribing a rectangle inside a circle, knowing how to work with perpendicular lines becomes essential.
  • In a rectangle, all angles are right angles.
  • The intersection of two perpendicular lines forms these right angles.
  • These lines help in defining the correct shape and positioning of the rectangle within the circle.
Using perpendicular lines properly ensures that the rectangle's sides remain straight and form the proper angles, essential for both aesthetic and structural integrity.

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Most popular questions from this chapter

a. Draw a circle. Place points \(A, B,\) and \(C\) on it in such positions that \(m \overline{A B}+m \overline{B C}\) does not equal \(m \overline{A C}\). b. Does your example in part (a) contradict Postulate \(16 ?\)

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

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

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; \(34^{\circ} \mathrm{S}\)
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