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