1. Circles
Chapter 12

2. Parts of a Circle A • M B C D E
Chord
diameter
radius
Circle: The set of all points in a plane that are a given distance from a given point in that plane.
Center: The middle of the circle – a circle is named by its center, the symbol of a circle looks like - סּM.
Radius: a segment that has one endpoint at the center and the other endpoint on the circle. The radius is ½ the length of the diameter. ALL RADII ARE CONGRUENT.

3. More Vocabulary A • M B C D E
Chord
diameter
radius
Chord: a segment that has its endpoints on the circle.
Diameter: a chord that passes through the center of the circle. The diameter is 2 times the radius.
Circumference: the distance around the circle. To find the circumference use:
Arcs: the space on the circle between the two points on the circle.

4. Tangent Lines
Tangent line: A line that intersects the circle at exactly one point is a tangent line to סּT. • T P A B
Point of Tangency: the point where the circle and the tangent line intersect.
Theorem 12-1: If a line is tangent to a circle, then the line is perpendicular to a radius drawn to the point of tangency.

5. Tangent Lines Continued• X P Q R
Theorem 12-3: Two segments tangent to a circle from the same point outside of the circle are congruent.

6. Central Angles
Central Angles: angles whose vertex is the center of the circle. • D E F G
37º
Theorem 12-4:
Within a circle or congruent circles:
Congruent central angles have congruent chords.
Congruent chords have congruent arcs.
Congruent arcs have congruent central angles.

7. Chords • Theorem 12-5: Within a circle or congruent circles
Chords equidistant from the center are congruent.
Congruent chords are equidistant from the center. • P A E B C F D

8. More About Chords • Theorem 12-6: Theorem 12-7: Q Theorem 12-8:
In a circle, a diameter that is perpendicular to a chord bisects the chord and its arc. • T V U R S Q
Theorem 12-7:
In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord.
Theorem 12-8:
In a circle, the perpendicular bisector of a chord contains the center of the circle.

9. Inscribed Angles
Inscribed Angle: An angle whose vertex is on the circle, and the sides are chords of the circle. • D A B C
Intercepted Arc: an arc of a circle having endpoints on the sides of an inscribed angle. AB

10. Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc. ● A B C
90°
45°
Corollaries:
Two inscribed angles that intercept
the same arc are congruent.
An inscribed angle in a semicircle is a right angle.
The opposite angles of a quadrilateral inscribed in a circle are supplementary. 1
38° ● 2 1 2 3 4

11. An angle formed by a tangent line and a chord.
The measure of angle formed by a tangent line and a chord is half the measure of the intercepted arc. ● C D B
65°
130°

12. Secant Line
A secant line is a line that intersects a circle at two points. ●
Theorem 12-11:
The measure of an angle formed by two lines that
Intersect inside a circle is half the sum of the measure of the intercepted arcs.
Intersect outside the circle is half the difference of the measures of the intercepted arcs. x° 1 y° 1 y° ● x°

13. Segment Length Theorems
3. (y + z)y = t2
1. a ● b = c ● d t y z ● a b c d ●
2. (w + x)w = (y + z)y w x y z ●

14. Equation of a Circle
An equation of a circle with the center (h, k) and radius r is (x – h)2 + (y – k)2 = r2.
Example: Center (5, 3) radius 4.
(x – 5)2 + (y – 3)2 = 16