CONCEPT PREVIEW Find the radius of each circle.

Verified step by step guidance

1. Identify the given information about the circle, such as the coordinates of the center and a point on the circle, or the length of a chord, diameter, or any other relevant measurement.
2. Recall the formula for the radius of a circle. If you know the center \((h, k)\) and a point on the circle \((x, y)\), use the distance formula to find the radius: \[r = \sqrt{(x - h)^2 + (y - k)^2}\]
3. If the problem provides the diameter \(d\), remember that the radius is half the diameter: \[r = \frac{d}{2}\]
4. If the problem involves trigonometric relationships (such as angles subtended by chords or arcs), use the appropriate trigonometric formulas to express the radius in terms of the given angles and lengths.
5. Substitute the known values into the chosen formula and simplify the expression to find the radius. Avoid calculating the final numeric value unless asked.

Key Concepts

Definition of a Circle and Radius

A circle is the set of all points equidistant from a fixed point called the center. The radius is the distance from the center to any point on the circle. Understanding this definition is essential to identify or calculate the radius when given information about the circle.

Using the Distance Formula

When the coordinates of the center and a point on the circle are known, the radius can be found using the distance formula: \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This formula calculates the length between two points in the coordinate plane.

Equation of a Circle

The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius. Recognizing or manipulating this equation helps in extracting the radius directly from the equation.