12.1: Tangent Lines
Congruent Circles: circles that have the same radius length
Diagram of Examples
Center of Circle:
Circle Name:
Radius:
Diameter:
Chord:
Secant:
Tangent to A Circle: a line in the plane that intersect a circle at one exact point
Point of Tangency: point at which the tangent line intersects the circle
Theorem 12.1: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of
tangency.
Theorem 12.2: In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle,
then the line is tangent to the circle.
Theorem 12.2: If two tangent segments to the same circle come from the same point outside the circle,
then the two tangent segments are congruent
Examples: Find the value of x in the following.
1)
2)
Examples: Find the radius of each circle.
3)
4)
5) Determine the perimeter of the polygon. Assume that all lines are tangent to the circle.
Examples: Determine if the line is tangent to the circle.
6)
7)
12.2: Chords and Arcs
Central Angle:
Minor Arc:
Major Arc:
Semicircle:
Diagram
A
Central Angle:
Minor Arc:
B
Major Arc:
C
Semicircle:
D
Theorem 12.4: In the same circle, or in congruent circles, two minor arcs are congruent if and only if
their corresponding chords are congruent.
Theorem 12.5: Central angles are congruent if and only if their chords are congruent.
Theorem 12.6: Chords are congruent if and only if their arcs are congruent.
Examples: Given that the circles are congruent, what can you conclude based on the figures.
1)
2)
Theorem 12.7: In the same circle, or congruent circles, two chords are congruent if and only if they are
equidistant from the center.
Examples: Find the value of x.
3)
4)
Theorem 12.8: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord
and its arc.
Examples: Find the value of x.
5)
6)
Theorem 12.9: If a diameter bisects a chord, then it’s perpendicular to the chord.
Theorem 12.10: The perpendicular bisector of a chord contains the center of the circle.
If you want to find the center, bisect 2 chords and find the point that they meet at.
12.3: Inscribed Angles
Inscribed Angle:
Intercepted Arc:
Diagram
Inscribed Angle and Intercepted Arc:
A
C
B
Theorem 12.11: Inscribed Angle Theorem
If an angle is inscribed in a circle, then the measure is half the measure of the intercepted arc.
Inscribed Polygon: all vertices of a polygon lie on the circle, the circle is drawn around
Circumscribed: when a circle is drawn about a figure
Corollary to 12.11: Two inscribed angles that intercept the same arc are congruent.
Corollary to 12.11: An angle inscribed in a semicircle is a right angle.
Corollary to 12.11: The opposite angles of an inscribed quadrilateral are supplementary (ADD TO 180)
Examples: Find the value of each variable.
1)
2)
3)
4)
5)
7)
6)
8)
Theorem 12.12: If a tangent and a chord intersect at a point on a circle, then the measure of each angle
formed is one-half the measure of its intercepted arc.
Examples: Find the value of the variable.
9)
10)
12.4: Angles Measures and Segment Lengths
Secant Segment: segment that extends through the circle
Theorem 12.13: If two chords or secants intersect inside the circle, then the measure of each angle
formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Example: Find the value of x.
Theorem 12.14: If a tangent and a secant, two tangents, or two secants intersect on the outside of the
circle, the measure of the angle formed is one-half the difference of the intercepted arcs.
Examples: Find the value of x.
Examples: Find the value of the numbered angle.
1)
2)
3)
4)
6)
5)
a(b) = c(d)
w(x + w) = y(z + y)
Examples: Find the value of the variable.
t2 = y(z + y)
12.5: Circles in the Coordinate Plane
Circle: set of all points in a plane that are equidistant from a given point known as the center of the circle
Equation of a Circle
(x – h)2 + (y – k)2 = r2
Center = (h, k)
Example: Write the standard equation of a circle given
1) centered at (3,15) that covers a radius of 7
Radius = r
2) center (2, 5); r =
Example: Identify the center and radius of the following circles.
3) (x – 3)2 + (y – 2)2 = 9
4) (x +2)2 + (y + 1)2 = 4
Example: Write the equation of the given circle
6) Write the equation for a circle
with center (1, -3) and passing through (2, 2).
You will need to use the distance formula.
y


x






7)
2
5) (x + 4)2 + (y - 2)2 = 3
Example: Graph the circles:
8) x2 + y2 = 16
9) (x-4)2 + (y + 3)2 = 4
y
y




x






x





10) Write an equation of a circle with diameter AB . The endpoints are given.
A(0, 0), B(6, 8)

12.6: Locus
Locus: set of all points in a plane that satisfies a given condition
Loci – plural of locus, pronounced “low-sigh”
Examples: Sketch each set of loci. Then describe each set.
1) points 1.5 cm from a point T
2) points 1 in. from PQ
3) points equidistant from the endpoints of AB
4) points that belong to a given angle or its interior
and are equidistant from the sides of the given angle
Sketching a Locus for Two Conditions
Example: Sketch the locus of points that are equidistant from X and Y and 2cm from the midpoint of XY .