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 quantities.
5. Substitute the known values into the chosen formula and simplify the expression to find the radius. Avoid calculating the final numeric value unless required.

Key Concepts

Definition of a Circle and Radius

A circle is a set of points equidistant from a fixed center point. The radius is the distance from the center to any point on the circle, and it is constant for a given circle. Understanding this helps identify what is being asked when finding the radius.

Using the Distance Formula

If the coordinates of the center and a point on the circle are given, 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 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 allows extraction of the radius by comparing terms.