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Unit 25

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

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Unit 25

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  1. Unit 25 CIRCLES

  2. 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

  3. B F • • A • C • • E • D 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

  4. 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

  5. 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

  6. 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:

  7. 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

  8. C D  O A B 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 • Solving the proportion, arc AB = 15.13 cm Ans

  9. 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

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

  11. C E B A D 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) 14 in (2 in) = (4 in)(EB) EB = 7 in Ans

  12. A • C • + • E • • B 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: • 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

  13. A B P C D 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 APD = ½ (98 – 40) = 29 Ans

  14. 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

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

  16. 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.

  17. B + D C A + A 1 B 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

  18. A B C E C P D F 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.

  19. 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

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

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