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Chapter 1: Problem 81

In Exercises 77-82, find the center and radius of the circle, and sketch its graph. \( (x-\frac{1}{2})^2 + (y-\frac{1}{2})^2 = \frac{9}{4} \)

Short Answer

Expert verified
The center of the circle is (0.5, 0.5) and its radius is 1.5.

Step by step solution

01

Identify the Center of the Circle

Looking at our equation \( (x-\frac{1}{2})^2 + (y-\frac{1}{2})^2 = \frac{9}{4} \). It is obvious that our 'h' and 'k' values are both \(\frac{1}{2}\). Thus, the center of our circle is at point (\(\frac{1}{2}\), \(\frac{1}{2}\)).
02

Calculate the Radius

The radius squared is given on the right-hand side of the equation, which is \(\frac{9}{4}\). To get the radius, we take the square root of this number. The square root of \(\frac{9}{4}\) is \(\frac{3}{2}\). Therefore, the radius of the circle is \(\frac{3}{2}\).
03

Sketch the Circle

Plot the center of the circle at (\(\frac{1}{2}\), \(\frac{1}{2}\)). Then, use the radius to draw the circle around this point. It should touch points \(\frac{3}{2}\) distance from the center, both horizontally and vertically.

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Key Concepts

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

Center of a Circle
The center of a circle is a pivotal point you need to identify when working with the equation of a circle. In its standard form, the equation of a circle is written as \[ (x-h)^2 + (y-k)^2 = r^2 \]where
  • \( (h, k) \) represents the center of the circle, and
  • \( r \) is the radius.

The values \( h \) and \( k \) are coordinates of the center. In the given equation \[ (x-\frac{1}{2})^2 + (y-\frac{1}{2})^2 = \frac{9}{4} \]you simply compare it to the standard form to identify the center as \( (\frac{1}{2}, \frac{1}{2}) \).
Knowing the center is essential for sketching and analyzing the circle's properties.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle's edge. It is a constant value for a particular circle. In the circle's equation \[ (x-h)^2 + (y-k)^2 = r^2 \]
The right-hand side is the square of the radius (\( r^2 \)).
  • To find the radius, take the square root of \( r^2 \).
  • For our specific case in this exercise, \( r^2 = \frac{9}{4} \).
  • The square root of \( \frac{9}{4} \) is \( \frac{3}{2} \).

This means the circle has a radius of \( \frac{3}{2} \). Understanding the radius is crucial as it determines the size of the circle when graphing or analyzing its extent in a coordinate plane.
Graphing Circles
Graphing circles requires both the center and the radius, as derived from the standard form equation. Once identified, you follow a straightforward approach:
  • First, plot the center of the circle on the coordinate plane. For our equation, it's at \( (\frac{1}{2}, \frac{1}{2}) \).
  • Next, measure the radius distance out from the center in all directions. In this case, it's \( \frac{3}{2} \) units.

Sketching the circle involves ensuring the entire circle remains equidistant from the center across all points. Consider using a compass or circular guide on graph paper to maintain accuracy. This visual representation makes it easier to analyze and understand geometric properties, like symmetry, location, and boundary relations within the plane.

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