*note: there are 360 degrees in a circle
If mFG = 80, find mFEG
mFEG = * Do not confuse the degree measure of an arc with the length of an arc. They are very different!
45
45
Measures and congruent are two different things. Measures are degrees (angles and arc measure). length Congruent means length or same 2cm
length size (lines and arc length)
3cm
Congruent Circles: Circles with congruent radii
Congruent Arcs: A 75 B
C
Arcs that are equal in measure AND in the same or congruent circles.
F
O
75
G
80
*Congruent means LENGTH not measure
80
11­2 Arcs and Chords
Defn: Chord­
Defn: Diameter­
d=2r (twice radius)
Thm 54: In a circle or congruent circles, congruent central angles have congruent chords.
Thm 55: congruent arcs have congruent chords.
Circumscribed: The outside figure
Inscribed: The inside figure
Ex: Triangle ABC is inscribed in Circle O.
Ex: Polygon ABCD is circumscribed about Circle O.
Common internal tangent
tangent intersects the line that joins the centers
Common external tangent
tangent does not intersect the line that joins the centers
Externally Tangent Circles
the only point of intersection is the point of tangency
Internally Tangent Circles
all of the points of one circle are
inside the other circle except the point of tangency
Tangent Segments ­ a segment of a tangent line where one of the endpoints is the point of tangency.
Q
P
Thm : Tangent segments drawn to a circle from an external point are congruent.
(can prove this using H­L)
Corollary : If two tangents are drawn to a circle from an exterior point, the line determined by that point and the center of the circle bisects the angle formed by the tangents.
11­5 Angles formed by Tangents, Chords, and Secants
Thm: The measure of an angle formed by a tangent and a chord intersecting at the point of tangency is equal to one­half the measure of the intercepted arc.
Thm : The measure of an angle formed by two chords intersecting within a circle is equal to one­half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Thm : The measure of an angle formed by a tangent and a secant, or two secants, or two tangents intersecting outside a circle is equal to one­half the difference of the measures of the intercepted arcs. (3 cases)
1. 2.
3.
Ex1: Triangle ABC is circumscribed about circle O, with D, E, and F as points of tangency for AB, BC, and CA, respectively. If AF=6 and EB=7, find the length of AB.
Ex2: Point P is 9 inches from the center of a circle whose radius measures 6 inches. Find the exact length of a tangent segment from P to the circle