91Ó°ÊÓ

The center of a circle on a coordinate plane is a crucial aspect that determines the circle's position. It is usually denoted by the coordinates \((a, b)\). These coordinates are directly derived from the equation of the circle given in standard form: \((x-a)^2 + (y-b)^2 = r^2\). In the equation, \(a\) and \(b\) represent the center of the circle.
To identify the center from an equation such as \(\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=\frac{9}{4}\), you can observe the values subtracted from \(x\) and \(y\) inside the parentheses.
The terms \(-\frac{1}{2}\) for \(x\) and \(-\frac{1}{2}\) for \(y\) indicate that the center of the circle is at \((\frac{1}{2}, \frac{1}{2})\).
This point, \((\frac{1}{2}, \frac{1}{2})\), acts as a fixed point around which the entire circle is evenly distributed on the coordinate plane.

The radius of a circle is a measure of how large or small the circle is, extending from its center to any point on its circumference. Simply put, it is the distance from the center of the circle to its edge.
In a circle's equation \((x-a)^2 + (y-b)^2 = r^2\), the \(r^2\) term represents the square of the radius. So, to find the radius, you need to take the square root of this term.
For the equation \(\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=\frac{9}{4}\), the square of the radius is \(\frac{9}{4}\).

• Square root of \(\frac{9}{4}\) is \(\frac{3}{2}\).

This tells us that the radius is \(\frac{3}{2}\) units. It shows all points on the circle are \(\frac{3}{2}\) units away from the center.

The coordinate plane is an essential tool for graphing shapes and solving geometric problems. It consists of two axes, the horizontal \(x\)-axis and the vertical \(y\)-axis, meeting at the origin \((0,0)\), which divides the plane into four quadrants.
In our exercise, the coordinate plane helps locate the circle's center and outline its radius for accurate graphing.

• Locate the center point \((\frac{1}{2}, \frac{1}{2})\) on the plane.
• From this center, use the radius \(\frac{3}{2}\) to mark points that distance away in all directions.

This method ensures that when you draw the circle, it is symmetrical around the center and maintains uniform distance (the radius) in every direction. Understanding how to utilize the coordinate plane is fundamental in graphing and visualizing the properties of the circle.