chapter 10 notes Circles

Section 10.1  Circles and Circumference

 

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.

 

 

 

 

 

 

Diametersegment across a circle through the center.

 

 

 

 

Chordsegment 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

 Secantline 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)

         

 

                 

 

 

 

 

Section 10.8 Equations of Circles

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