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Explore the fundamentals of circle geometry, from the definition of a circle with its radius, diameter, and circumference, to complex concepts like sectors and tangents. Understand the standard and general forms of circle equations in coordinate geometry, and discover key theorems that reveal the relationships between angles and arcs. Gain insights into constructing equations from circle graphs and transforming general circle equations.
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A circle is a two-dimensional shape with all points at a constant distance from a fixed point
Radius
The radius is the distance from the center to any point on the circle
Diameter
The diameter is the longest straight line that can be drawn within the circle, passing through the center
Circumference
The circumference is the total distance around the circle, calculated using the formula C = 2πr or C = πd
The equation of a circle on a coordinate plane is (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius
Circle theorems describe the relationships between angles, lines, and arcs within or around a circle
Congruence of angles
Angles subtended by the same arc in the same segment are congruent
Sum of opposite angles
The sum of opposite angles in a cyclic quadrilateral is 180 degrees
Tangent and radius relationship
The angle between a tangent and a radius is perpendicular at the point of contact
The central angle subtended by an arc is twice the angle subtended at the circumference
To derive the equation of a circle from its graph, one must identify the coordinates of the center and the length of the radius
The Pythagorean theorem can be used to find the radius by measuring the distance between the center and any point on the circumference
The general form of a circle equation, x^2 + y^2 + Dx + Ey + F = 0, can be converted to the standard form by completing the square for the x and y terms