# G.12 and *G.12 - Equation of a Circle

G.12    The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.

G.12  The student will solve problems involving equations of circles.

Bloom's Level:  Analyze

### BIG IDEAS

• I can determine the percentage of corn fields being watered by circular irrigation sprinklers, how many miles a tire can travel before wearing off its tread, and the best spot for a soccer player to stand on the sideline to make a goal.
• I will remember that when given the general form of a circle equation I can easily convert it to center-radius formby completing the square.

### UNDERSTANDING THE STANDARD

• A circle is a locus of points equidistant from a given point, the center.
• Standard form for the equation of a circle is (x - h)2 + (y - k)2 = r2, where the coordinates of the center of the circle are (h,k) and r is the length of the radius.
• The circle is a conic section.

·  A circle is a locus of points equidistant from a given point, the center.

·  The distance between any point on the circle and the center is the length of the radius.

·  Standard form for the equation of a circle is (xh)2 + (yk)2 = r2, where the coordinates of the center of the circle are (h, k) and r is the length of the radius.

·  The equation of a circle gives the coordinates of every point, (x, y), on the circle.

·  The midpoint formula and distance formula are important when determining the equation of a circle.

·  The equation of a circle with a given center and radius can be derived using the Pythagorean Theorem.

·  The midpoint of the diameter is the center of the circle.

### ESSENTIALS

G.121  Identify the center, radius, and diameter of a circle from a given standard equation.

G.124  Given the equation of a circle in standard form, identify the coordinates of the center and find the radius of the circle.

·  G.121  Given a graph or the equation of a circle in standard form, identify the coordinates of the center of the circle.

·  G.123  Given a graph or the equation of a circle in standard form, identify the length of the radius or diameter of the circle.

G.122  Use the distance formula to find the radius of a circle.

·  G.125  Given the coordinates of the center and the coordinates of a point on the circle, determine the length of the radius or diameter of the circle.

G.123  Given the coordinates of the center and radius of the circle, identify a point on the circle.

·  G.126  Given the coordinates of the center and length of the radius of a circle, identify the coordinates of a point(s) on the circle.

G.125  Given the coordinates of the endpoints of a diameter, find the equation of the circle.

·  G.124  Given the coordinates of the endpoints of the diameter of a circle, determine the length of the radius or diameter of the circle.

·  G.122  Given the coordinates of the endpoints of a diameter of a circle, determine the coordinates of the center of the circle.

G.126  Given the coordinates of the center and a point on the circle, find the equation of the circle.

·  G.127  Determine the equation of a circle given:

­  a graph of a circle with a center with coordinates that are integers;

­  coordinates of the center and a point on the circle;

­  coordinates of the center and the length of the radius or diameter; or

­  coordinates of the endpoints of a diameter.

G.127  Recognize that the equation of a circle of given center and radius is derived using the Pythagorean Theorem.

### KEY VOCABULARY

circle, center, point, equation, equation of circle, (x,y), (h,k), radius, diameter, standard equation, distance formula, locus of points, coordinates, identify, endpoint, Pythagorean Theorem, equidistant, conic section

Updated: Feb 22, 2018