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Chapter 10: Problem 12

Draw a circle. Circumscribe a square about the circle.

Short Answer

Expert verified
The process involves drawing a circle first and then a square around it, with the circle touching all four corners of the square. The square is the circumscribed polygon.

Step by step solution

01

Draw a Circle

Start by drawing a circle. You can do this using a compass or a circular object. Make sure it's a full and complete circle.
02

Draw the Square

To inscribe a square about the circle, you will need to draw it outside the circle. Start at the top of the circle and draw a straight line horizontally out on either side of the circle till you feel you have enough space for the square. Do the same at the bottom. These will be the top and bottom of your square.
03

Complete the Square

Next, draw two lines vertically down from the ends of the top line until they reach the bottom line to complete the square. The circle should now be inscribed within the square.

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Key Concepts

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

Geometry
Geometry is a fascinating area of mathematics focused on the study of shapes, sizes, relative positions of figures, and the properties of space. An ancient discipline, it's the cornerstone of modern mathematical thinking and plays a crucial role in our everyday lives, engineering, architecture, and more. In the context of our exercise, geometry allows us to understand the properties and relations between a circle and a square, two fundamental shapes. Circumscribing a square around a circle is an exercise that combines understanding the dimensional properties of these shapes and applying geometric concepts to create a square that perfectly encompasses the circle.
Circle
A circle is a round plane figure whose boundary (the circumference) is equidistant from a central point. Every point on the boundary is the same distance from the center, which is known as the radius. Circles are ubiquitous in geometry, and their properties have numerous applications in real-world scenarios. In our scenario, the circle is the starting shape around which a square must be circumscribed, meaning the circle will touch the square at exactly four points along the square's sides. Understanding the circle's radius and center point is key for accurately constructing the square around it.
Square
A square is a type of rectangle that has four equal sides and four right angles. It's one of the simplest forms of polygons but also a powerful one because of its symmetry and uniformity. In the problem at hand, creating a square around a circle involves precise construction to ensure that all four sides are equal and the sides are perpendicular to one another. To circumscribe a circle within a square means that the diameter of the circle is equal to the side length of the square, responding to a particular balance between the two shapes within the realm of geometry.
Compass Construction
Compass construction is a means of constructing figures using only a compass and straightedge (no measurements). This method goes back to ancient times and is still taught today because it emphasizes accuracy and process in geometric constructions. In order to circumscribe a square around a circle, the compass is used to draw the initial circle with a defined radius. From there, we extend lines from the circle's boundary to create the square. The process requires the constructive alignment of the square's sides with the tangential points on the circle, showcasing the ingenuity and precision of compass constructions in geometry.

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Most popular questions from this chapter

Draw diagrams to show the possibilities with regard to points in a plane. Given a point \(A\) and two parallel lines \(j\) and \(k\), what is the locus of points \(30 \mathrm{cm}\) from \(A\) and equidistant from \(j\) and \(k ?\)

Construct an angle with the indicated measure. $$45$$

Refer to plane figures. Begin each part of this exercise by drawing any \(\overline{C D}\). Then construct the locus of points \(P\) that. meet the given condition. a. \(\angle C D P\) is a right angle. b. \(\angle C P D \text { is a right angle. (Hint: See Classroom Exercise } 2 .)\)

Construct a large right triangle. Construct the circumscribed circle.

Draw \(\overline{C D}\) about \(20 \mathrm{cm}\) long. Construct a triangle whose perimeter is equal to \(C D\) and whose sides are in the ratio 2: 2: 3 .
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