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Chapter 14: Problem 19

Given a circle Q with a radius of 9 , find the locus of points 9 units from the circle Q.

Short Answer

Expert verified
The locus of points 9 units from circle Q is another circle, with the same center as Q, and radius of 18 units.

Step by step solution

01

Identify Known Variables

First, identify the circle Q with a given radius of 9 units. The purpose is to find the locus of points 9 units from the circle Q.
02

Understand the Locus Concept

Locus is defined as the set of points that share a property or fulfill a condition. In this case, we seek the locus of points which are 9 units from any point on circle Q.
03

Determine the Locus

As the points to be located are 9 units from all points on circle Q, which is essentially the distance of a radius away, we can conclude that the locus forms a circle itself. This new circle is centered at the same point as circle Q but has a larger radius, which is exactly the radius of Q plus the given 9 units distance from circle Q.
04

Calculate the New Radius

As the radius of Circle Q is 9 units, the radius of the locus circle, let's call it Circle L, would be the radius of Q plus the distance of our points from circle Q. Therefore the radius of circle L is: \(9 units + 9 units = 18 units\)

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Key Concepts

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

Circle and Radius
A circle is a perfect round shape where every point on its boundary is equidistant from the center point. This distance from the center of the circle to any point on the edge is known as the radius. The radius is a fundamental aspect of a circle. In the given exercise, the circle Q has a radius of 9 units. This means each point on the perimeter of the circle is 9 units away from the center of circle Q.
  • The radius serves as a crucial measure to understand and describe the size of a circle.
  • This measure is also fundamental in many geometric calculations and constructions involving circles.
  • Knowing the radius is essential to finding other properties of the circle, like its circumference and area.
Geometric Construction
Geometric construction refers to the process of drawing geometric figures using only a compass and straightedge, without the need for measuring them. In this context, it helps us understand how to create new shapes based on existing ones.
  • To determine the locus of points that are 9 units from circle Q, we essentially perform a geometric construction.
  • The construction involves creating a new circle which remains uniformly 9 units away from every point on circle Q.
  • This requires understanding the nature of circles and the ability to visualize where all these points would lie if traced around circle Q.
Through geometric construction, we can build complex figures by understanding simple properties, such as maintaining a constant distance, which is a key element of our exercise when determining the locus.
Locus of Points
The term 'locus' in geometry refers to a set of points that satisfy a particular condition or a set of conditions. In the exercise, we are tasked with finding the locus of points that are exactly 9 units away from every point on circle Q.
  • In practical terms, this means we need to locate all points that are at a fixed distance from each point on another shape (circle Q in this case).
  • Here, since each point on the locus is a fixed 9 units away from every point on circle Q, these points will form a circle of their own.
  • The new circle has the same center as circle Q but extends further out, creating a perfect illustration of locus.
The concept of locus helps solve problems where you need to find or construct a path or region defined by a specific property, such as distance, like in this exercise where we found our locus as a new circle with a radius of 18 units, centered on circle Q.

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Most popular questions from this chapter

Draw a sketch and write a description of each locus. The locus of points equidistant from two given points

a The locus of points equidistant from the vertices of a triangle is the point of intersection of the \(\underline{?}\) of the triangle. b The locus of points equidistant from the sides of a triangle is the point of intersection of the \(\underline{?}\) of the triangle.

Find the locus of points equidistant from two concentric circles and on a diameter of the larger circle.

Given \(\triangle \mathrm{PQR}\), trisect \(\overline{\mathrm{QR}}\).

Draw a sketch and write a description of each locus. The locus of points equidistant from two given concentric circles (If the radii of the circles are 3 and \(8,\) what is the size of the locus?)
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