2.5
To determine the radius of a dartboard's largest scoring ring, it can be modeled as a circle.
A circle consists of all points that are the same fixed distance, called the radius, from a central point known as the center.
The equation of a circle comes from the distance formula, which is based on the Pythagorean Theorem. It measures the distance between the center and any point on the circle. As point P moves along the edge, its distance from the center stays constant. Squaring both sides removes the square root and produces the standard form of the equation.
With the center at the origin, the equation reduces to the sum of the squares of the x- and y-coordinates.
To graph a circle from its equation, the coordinates of the center are picked from the standard form, then the radius is identified.
Conversely, given a graph of a circle, locating the center and measuring the radius allows direct substitution into the standard form to get the equation.
If the equation is not in standard form, grouping terms and completing the square reveals the circle’s center and radius.
A circle in the coordinate plane is defined as the set of all points that lie at a constant distance, known as the radius, from a fixed point called the center. This relationship is captured using the distance formula. For a point (x, y) on the circle and a center (h, k), the distance between them equals the radius r. By squaring both sides of the distance formula, the equation of the circle is written in standard form:
Constructing the Equation from Geometric Information
If the center and the radius of a circle are known, the equation can be directly formed using substitution. Consider a circle with its center at (1, 2) and radius 4. Substituting into the standard form yields:
This equation represents all points (x, y) that lie 4 units from the point (1, 2). This simple approach is often sufficient for constructing equations when the geometric characteristics of the circle are explicitly given.
Identifying the Circle from a General Equation
When the circle’s equation is not presented in standard form, completing the square helps in identifying its center and radius. Take the equation
Group and complete the square:
This reveals the center at (3, 2) and a radius of 4. Rewriting equations in standard form is essential for graphing and analyzing circles effectively.
To determine the radius of a dartboard's largest scoring ring, it can be modeled as a circle.
A circle consists of all points that are the same fixed distance, called the radius, from a central point known as the center.
The equation of a circle comes from the distance formula, which is based on the Pythagorean Theorem. It measures the distance between the center and any point on the circle. As point P moves along the edge, its distance from the center stays constant. Squaring both sides removes the square root and produces the standard form of the equation.
With the center at the origin, the equation reduces to the sum of the squares of the x- and y-coordinates.
To graph a circle from its equation, the coordinates of the center are picked from the standard form, then the radius is identified.
Conversely, given a graph of a circle, locating the center and measuring the radius allows direct substitution into the standard form to get the equation.
If the equation is not in standard form, grouping terms and completing the square reveals the circle’s center and radius.
From Chapter 2:
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