Circles and Relationships
Circumference of a circle
2πr or πd
Area of a circle
A=πr²
If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.
If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are ___________________.
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and the arc.
In a circle, if a diameter (or radius) is perpendicular to a chord, then it _____________________________________.
Within a circle or in congruent circles, chords equidistant from the center(s) are congruent.
Within a circle or in congruent circles, chords ________________________ are ___________________
Within a circle or in congruent circles, congruent chords have congruent arcs.
Within a circle or in congruent circles, congruent chords have ________________________.
tangent line
a line that is in the same plane as a circle and intersects the circle at exactly one point
sector of a circle
a region bounded by two radii of the circle and their intercepted arc
sector
a region inside a circle bounded by two radii of the circle and their intercepted arc
diameter
a segment that passes through the center of the circle and whose endpoints are on the circle
radius
a segment whose endpoints are the center and any point on the circle; the distance from the center of a circle to any point on the circle
secant line
A line that intersects a circle at two points
segment of a circle
A region inside a circle bounded by a chord and an arc
chord
A segment whose endpoints lie on a circle
major arc
An arc of a circle whose points are on or in the exterior of a central angle
ϴ/360 * 2πr (where ϴ is the angle, and r is radius)
How do you find arc length of a sector?
A(sector)-A(triangle) A(segment)
How do you find the area of a segment?
d=√(x2−x1)²+(y2−y1)²
How do you find the radius if you have the center and point on the circle?
m<2= 1/2(m[arc]AC+ m[arc]BD)
How do you solve if there are TWO secants or chords that intersect in the interior of the circle?
m<ABC=1/2m(arc)AB
How do you solve tangent and secant angles with a vertex on the circle?
If a line in the plane of a circle is perpendicular to a radius at its endpoints on the circle, then the line is tangent to the circle.
If a line in the plane of a circle is perpendicular to a radius at its endpoints on the circle, then the line is ______________________________.
If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.
If a line is tangent to a circle, then the line is ______________________________.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc
A=πr²
What is the area of a circle?
-AE * EB = CE * ED -Multiply the two sides of each chord and set them equal to other
What is the chord-chord product theorem?
(x-h)²+(y-k)²=r²
What is the equation of a circle?
1. If two angles intercept the same arc, then the angles are congruent.
What is the first theorem for inscribed angles?
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
What is the fourth theorem for inscribed angles?
-AE * BE = CE * DE -Add the two sides of each secant -Multiply total secant by outside portion and set the secants equal to each other
What is the secant-secant product theorem?
-AC * BC = DC² -Add two sides of secant and multiply by outside portion and set it equal to the tangent squared
What is the secant-tangent product theorem?
2. If two angles intercept congruent arcs, then the angles are congruent.
What is the second theorem for inscribed angles?
3. If an inscribed angle is a right angle, then the intercepted arc is a semi-circle.
What is the third theorem for inscribed angles?
-Vertex in circle
Which angle relationships do you ADD?
-Vertex on circle (multiply by 2, or divide by 1/2)
Which angle relationships do you MULTIPLY by 2?
-Vertex outside circle (secant and tangent, 2 secants, 2 tangents)
Which angle relationships do you SUBTRACT?
Within a circle or in congruent circles, congruent central angles have congruent chords.
Within a _____________________, congruent central angles have ___________________________.
inscribed angle
an angle whose vertex is on a circle and whose sides contain chords of the circle
central angle
an angle whose vertex is the center of the circle
semi-circle
an arc of a circle whose endpoints lie on a diameter
minor arc
an arc of a circle whose points are on or in the interior of a central angle
intercepted arc
an arc that consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between the endpoints
circumference
the distance around the circle
center
the point inside a circle that is the same distance from every point on the circle
point of tangency
the point of intersection of a circle or sphere with a tangent line or place
circle
the set of points in a plane that are a fixed distance from a given point called the center of the circle
