Unit 25 CIRCLES

DEFINITIONS circle - closed curve in which every point on the curve is equally distant from a fixed point called the center circumference - the length of the curved line that forms the circle chord - a straight line segment that joins two joints on the circle diameter - a chord that passes through the center of a circle radius - a straight line segment that connects the center of a circle with a point on the circle

DEFINITIONS (Cont) arc - that part of a circle between any two points on the circle. FB is an arc tangent - a straight line that touches the circle at only one point. DE is a tangent line A secant - a straight line passing through a circle and intersecting the circle at two points. AC is a secant line A • B C D E F

DEFINITIONS (Cont) segment - a figure formed by an arc and the chord joining the end points of the arc sector - a figure formed by two radii and the arc intercepted by the radii central angle - an angle whose vertex is at the center of the circle and whose sides are radii inscribed angle - an angle in a circle whose vertex is on the circle and whose sides are chords

CIRCUMFERENCE The circumference of a circle is equal to  times the diameter or 2 times the radius C = d or C = 2r Find the circumference of a circle whose diameter is 3 ft. Round your answer to three significant digits C = π(3) = 9.42 ft Ans

ARC LENGTH The length of an arc equals the ratio of the number of degrees of the arc to 360° times the circumference Determine the length of a 30° arc on a circle with a radius of 5 m:

CIRCLE POSTULATES In the same circle or in equal circles, equal chords subtend (cut off) equal arcs (arcs of equal length) In the same circle or in congruent circles, equal central angles subtend (cut off) equal arcs In the same circle or congruent circles, two central angles have the same ratio as the arcs that are subtended (cut off) by the angles A diameter perpendicular to a chord bisects the chord and the arcs subtended by the chord; the perpendicular bisector of a chord passes through the center of the circle

POSTULATE EXAMPLE Determine the length of arc AB is the figure below given that CD is 24 cm, COD = 92, and AOB = 58 The postulate “In the same circle, two central angles have the same ratio as the arcs that are subtended by the angles” applies here  O C D A B Solving the proportion, arc AB = 15.13 cm Ans

CIRCLE TANGENTS AND CHORD SEGMENTS A line perpendicular to a radius at its extremity is tangent to the circle; a tangent to a circle is perpendicular to the radius at the tangent point Two tangents drawn to a circle from a point outside the circle are equal and make equal angles with the line joining the point to the center If two chords intersect inside a circle, the product of the two segments of one chord is equal to the product of the two segments of the other chord

TANGENT AND CHORD EXAMPLES Find the value of APO in the figure below, given that APB = 84: • • P + A O B APO = ½ APB. Thus, APO = 42 Ans

TANGENT AND CHORD EXAMPLES Determine length EB in the figure given below, given that AE = 4 in, CE = 14 in, and ED = 2 in: (CE)(ED) = (AE)(EB) C B E A D 14 in (2 in) = (4 in)(EB) EB = 7 in Ans

ANGLES INSIDE A CIRCLE An angle formed by two chords that intersect within a circle is measured by one half the sum of its two intercepted arcs An inscribed angle is measured by one half its intercepted arc An angle formed by a tangent and a chord at the tangent point is measured by one half its intercepted arc Find the length of arc AEB in the figure below, given that CAB = 38: + • C A B E CA is tangent to AB so CAB is one half of arc AEB or, in other words, arc AEB = 2CAB  Thus, arc AEB = 76° Ans

ANGLES OUTSIDE A CIRCLE An angle formed outside a circle by two secants, two tangents, or a secant and a tangent is measured by one half the difference of the intercepted arcs Determine APD in the figure below, given that arc AD = 98 and arc BC = 40: APD is equal to one half the difference of arc AD and arc BC P A B C D APD = ½ (98 – 40) = 29 Ans

INTERNALLY AND EXTERNALLY TANGENT CIRCLES Two circles are internally tangent if both are on the same side of the common tangent line Two circles are externally tangent if the circles are on opposite sides of the common tangent line If two circles are either internally or externally tangent, a line connecting the centers of the circles passes through the point of tangency and is perpendicular to the tangent line

PRACTICE PROBLEMS Identify the central and inscribed angle in the circle below: Define each of the following: a. Tangent b. Arc c. Chord A B C D E F

PRACTICE PROBLEMS (Cont) Determine the circumference of a circle with a radius of 2.5 inches. Determine the arc length of a circle with a 5 m radius and a 50° arc.

PRACTICE PROBLEMS (Cont) Determine DB and arc ACB in the figure at right, given that AB = .6 m and arc AC = .4 m. Refer to the figure at right. The circumference of the circle is 110mm. Determine these values: The length of arc AB when 1 = 42° 1 when arc AB = 42 mm + D B C A + A B 1

PRACTICE PROBLEMS (Cont) Find the value of ABC in the figure below, given that arc AC = 100°. Find the number of degrees in arc DE in the figure below, given that arc CF = 106° and that EPD = 74. A B C E D P C F

PROBLEM ANSWER KEY Central angle = ACB, inscribed angle = EDF a. A straight line that touches the circle at only one point That part of a circle between any two points on the circle A straight line segment that joins two joints on the circle 15.71 inches 4.363 meters

PROBLEM ANSWER KEY (Cont) a. DB = 0.3m b. Arc ACB = 0.8m a. 12.83mm b. 137.45° 50° 42°