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Chapter 2: Problem 71

Explain how the center and radius of a circle can be used to graph the circle.

Short Answer

Expert verified
Plot the center at \(h, k\) and use the radius \(r\) to draw a circle around it.

Step by step solution

01

Identify the Center of the Circle

The center of the circle is given by a point \(h, k\). This point will serve as a reference for placing the circle on a graph. Mark this point on the coordinate plane.
02

Determine the Radius

The radius of the circle is the distance from the center to any point on the circle. Let the radius be \(r\).
03

Plot the Center

First, place a dot on the graph at the coordinates of the center \(h, k\).
04

Measure the Radius

Using a ruler or the graph's scale, measure a distance of \(r\) units from the center in all directions (up, down, left, right, and diagonally).
05

Draw the Circle

With the radius measured, use a compass or steady hand to draw a smooth curve connecting these points around the center, forming the circle.

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Key Concepts

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

Center of the Circle
The center of a circle is a crucial point in the graphing process. It is identified by the coordinates \(h, k\). This point is your main reference when you start drawing the circle on a coordinate plane.

To mark this point on the coordinate plane:
  • Locate the x-coordinate (h) on the x-axis.
  • Locate the y-coordinate (k) on the y-axis.
  • Where these two coordinates intersect is the center of your circle.

Always ensure that you accurately plot this point, as it affects the shape and position of your circle. The center is your starting point for measuring the radius and drawing the circle's boundary.
Radius
The radius is the distance from the center to any point on the circle. It is usually denoted by \(r\). Understanding the radius is essential as it helps determine the size of the circle.

Here鈥檚 how you can correctly use the radius:
  • From the center \(h, k\), measure a distance of \(r\) units in all directions: up, down, left, right, and diagonally.
  • Use a ruler or the graph's scale to keep the measurements precise.
  • Mark these points to help guide the shape of your circle.

The radius ensures that every point on the circle's edge is equidistant from the center. This constant distance is what forms the perfect round shape of a circle.
Coordinate Plane
The coordinate plane is a 2-dimensional space defined by an x-axis (horizontal) and a y-axis (vertical). It is crucial for plotting any geometric shape, including circles.

To effectively use the coordinate plane in graphing circles:
  • Understand the layout of the coordinate plane: positive and negative values on both axes.
  • Locate the center \(h, k\) on this plane accurately.
  • Measure the radius correctly using the plane鈥檚 scale.

The coordinate plane provides a structured, precise framework for plotting shapes. Familiarity with navigating around the plane will make graphing circles, and other figures, a straightforward task.

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