radius

1. Circle: set of all points in a plane equidistant from a fixed point called the center. secant diameter radius chord tangent Circle 4.1

2. Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at the point of tangency. Theorem 2 : Tangent segments from a common external point are congruent. R 32 Q 3x + 5 S 50 r C r 70 32 = 3x + 5 27 = 3x 9 = x B r2 + 702 = (r + 50)2 r2 + 4900 = r2 + 100r + 2500 2400 = 100r 24 = r Properties of Tangents 4.2

3. (1). In the same circle, or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. C B AB  CD if and only if D T S A (2). If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Q Since SQ  TR and SQ bisects TR, SQ is a diameter of the circle. R Properties of Chords 4.3

4. (3). If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If EG is a diameter and TR  DF, then HD  HF and GD  GF. F E H G (4). In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. D C G if and only if FE  GE D E A F B Properties of Chords 4.3

5. (1). The measure of a central angle is equal to the measure of its intercepted arc. A If G is the center of the circle and mAGB = 100o, then mAB = 100o. G B (2). The measure of an inscribed angle is one half the measure of its intercepted arc. T If R is a point on the circle and mTRS = 60o, then mTS = 120o. R S Central Angles & Inscribed Angles 4.4

6. (1). An angle inscribed in a semicircle is a right angle. C If BC is a diameter of the circle then mCAB = 90o. A B (2). A quadrilateral can be inscribed in a circle, if and only if opposite angles are supplementary. xo + 88o = 180o and yo + 100o = 180o x = 92o y = 80o 88° 100° y° x° Inscribed Angles 4.4

7. (2). The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs. (1). The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. Y X 80o 1 60o Z W D E 100o 40o A 1 B C Angles of a Circle 4.5

8. (2). The rule for finding segment lengths formed by two secants or a secant and a tangent is(outside)(whole) = (outside)(whole). 3 6 (1). The rule for finding segment lengths formed by two chords is (part)(whole) = (part)(whole). x 10 7 5 x 10 Circles and Segments 4.6

9. (2). Arc Length: In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360o. (1). Circumference: C = 2 r or C =  d A Example: 100o 8 B Circumference and Arc Length 4.7

10. (2). The formula for the Area of a Sectoris given by: (1). Area of a Circle: A = r2 A B Example: Let x represent the are of sector AB. 40o 8 C Area of a Circle, Area of Sector 4.8

11. (2). Volume: (1). Surface Area: A = 4r2 Example: Find the surface area and volume of a sphere whose diameter measures 14 cm. Surface Area and Volume of Sphere 4.9