chapter 10 notes Circles

**Circle: **set of
** all points** in a plane that are **
equidistant** from a given point (center).

**Radius: ****segment** from the center to a point on the
circle.

--congruent circles have congruent radii.

--all radii in a given circle are congruent.

**Diameter**:
**segment** across a circle
** through the center**.

**Chord**:
**segment
** whose endpoints are ** on** the circle.

**Circumference**:
the **distance around** a circle

**pi**:
the ratio of Circumference to diameter

__Section 10.2 Angles and Arcs__

**Central Angle**—an angle whose vertex is the center
of the circle.

**Minor Arc**—An arc on a circle with a central angle
less than 180 degrees.

**Major Arc**—An arc on a circle with a central angle
greater than 180 degrees.

**Semicircle**—An arc on a circle with a central angle
of exactly 180 degrees.

**The measure of an arc is equal to the measure of its
central angle.
**

Th 10.1: In the same or congruent circles, two arcs are congruent if and only if their central angles are congruent

Post 10.1: (**Arc
Addition Postulate**): the measure of an arc formed by two
adjacent arcs is the sum of the measures of the two arcs

**Arc Length**:
the length (using linear measurement scale) of a particular arc by treating it
as part of the full circumference of the circle

or

A = Arc Measure (in degrees)

= Arc length (in linear units...ft, cm,
in...)

= Diameter of the circle

__Section 10.3 Arcs and Chords__

**Th 10.2**: Two minor arcs in the same circle
(or congruent circles) are congruent if and only if their corresponding chords are
congruent.

**Th 10.3**: If a diameter (or radius) is perpendicular to a chord, then it bisects the
chord and its arc.

Inscribed Polygons and Circumscribed Circles

**Th 10.4**: Two chords of a circle are
congruent if and only if they are equidistant from the
center.

__Section 10.4 Inscribed Angles__

**Inscribed Angle**—an angle whose vertex is on a
circle and whose sides are chords.

**Intercepted Arc**—an arc on the interior of an
inscribed angle.

Th 10.5: (**Inscribed
Angle Theorem**): If an angle is inscribed in a circle, then the
measure of the intercepted arc is twice the value of the measure of the
inscribed angle

Th 10.6: If two inscribed angles of a circle intercept congruent arcs (or the same arc), then the angles are congruent

Th 10.7: If an inscribed angle intercepts a semicircle, the angle is a right angle

Th 10.8: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

__Section 10.5 Tangents__

**Tangent**—a
** line** that intersects a circle in ** exactly
one** point.

**Point of Tangency**—the point where a tangent line
intersects a circle.

Th 10.9: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency

Th 10.10: If a line is perpendicular to a radius at its endpoint on a circle, then the line is tangent to the circle

Th 10.11: If two segments from the same exterior point are tangent to a circle, then they are congruent.

__Section 10.6 Secants, Tangents, and
Angle Measures__

**Secant**—**line** that intersects a circle in
** two**
points.

Th 10.12: If two secants (or
chords) intersect in the interior of a circle, then the measure of an angle
formed is 1/2 the sum of the measure of the arcs intercepted by the angle and
its vertical angle (*the angle is the average of the intercepted arcs*)

**Angle = 1/2 * (Arc1 +
Arc2)**

Th 10.13: If a secant and a tangent intersect at a point of tangency, then the measure of the angle is 1/2 the measure of the intercepted arc

**Angle = 1/2 *
intercepted arc**

Th 10.14: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle is 1/2 * the difference in the intercepted arcs.

*Angle = 1/2 * (Arc1 -
Arc 2)*

Standard Equation of a Circle:

Center (h,k)

Radius r

**Examples:**

**Write an equation for the circle with center (3, -2) and
radius=5**

**Write an equation for the circle with center (-1, 4) and
diameter=8**

**Find the center and radius of a circle with equation (x-3)^2 +
(y+5)^2 = 36**

**Write an equation for the circle with center (-3, 9) and a
radius with endpoint (1,9)**

**Write an equation for the circle whose diameter has endpoints
(3, 4) and (3, -6)**

**Graph the circle (x+2)^2 + (y)^2 = 9**

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