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Chapter 7: Problem 13

Sketch and describe each locus in the plane. Find the locus of points that are at a distance of 3 in. from a fixed point \(O .\)

Short Answer

Expert verified
The locus is a circle centered at point O with a radius of 3 inches.

Step by step solution

01

Identify the Nature of the Locus

The problem requires finding all points at a constant distance from a given fixed point (point O). Loci involving a fixed distance from a point typically describe a circle in a plane.
02

Understand the Components of a Circle

A circle is defined as the collection of all points in a plane that are equidistant from a central point. The central point is the fixed point O, and the radius of the circle is the distance, which is 3 inches here.
03

Establish the Definition of the Locus as a Circle

Using the definition of a circle, we know the center is point O, and each point on the locus is 3 inches away from O. Thus, the locus is a circle with point O as its center and a radius of 3 inches.
04

Illustrate the Locus

Sketch a circle on the plane with point O as its center. Use a compass or any other tool ensuring every point on this circle is exactly 3 inches away from point O.
05

Describe the Locus

The locus of points is a circle centered at the fixed point O with a radius of 3 inches. Every point on this circle maintains a distance of 3 inches from O.

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Key Concepts

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

circle
In geometry, a circle plays a major role as a fundamental shape. When we think of a circle, imagine a perfect round shape in a plane where every point along its boundary is equidistant from a specific center point. This characteristic makes a circle distinct from other geometric shapes.
  • **Center:** The central point inside the circle from which all outer points are equally distant.
  • **Boundary:** The curved line that forms the perimeter of the circle. It's where the distance is measured consistently from the center.
The circle is considered a locus of points, meaning it is a set of points that share a specific property鈥攊n this case, being equidistant from the center point. This property gives rise to the circle's defining equation in a coordinate plane: \[ (x - h)^2 + (y - k)^2 = r^2 \]Where \(h, k\) represents the center of the circle, and \(r\) is the radius.
radius
The radius of a circle is a key concept that defines the space it occupies. Think of the radius as the invisible stick that connects the circle's center to its perimeter. It's a straight line segment that represents consistent distance.Factors of the Radius:
  • **Length:** Determines the size of the circle. A larger radius results in a larger circle.
  • **Constancy:** Every point on the edge of the circle is the exact same distance from the center, which proves invaluable in geometry.
When they ask for the locus of points at a certain distance from a fixed point, like the exercise does, they imply that these points form a circle where the radius equals that distance. In our example, every point on the circle is 3 inches away from the fixed point, \(O\), meaning the radius of the circle is 3 inches.
fixed point
A fixed point in geometry serves as a reference or starting point, around which specific calculations or constructions are made. In many cases, a fixed point is the anchor for defining various geometric figures, particularly circles.What is a Fixed Point?
  • **Definition:** A location in a plane that stays constant and does not move.
  • **Role in a Circle:** Acts as the center of the circle, from which all points on the circle's boundary are equidistant.
In the specific exercise discussed, the fixed point is labeled \(O\), and is the center of the circle. It serves as the focal reference from which radius measurements are consistently taken to construct the circle. By defining a fixed point, geometric structures like circles can be precisely delineated and understood.

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Most popular questions from this chapter

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Sketch and describe each locus in the plane. Find the locus of the midpoints of the radii of a circle \(O\) that has a radius of length \(8 \mathrm{cm}\).
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