Geometry Unit 10 Notes Circles. Syllabus Objective: The student will differentiate among the terms relating to a circle.


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1 Geometry Unit 0 Notes ircles Syllabus Objective: 0.  The student will differentiate among the terms relating to a circle. ircle the set of all points in a plane that are equidistant from a given point, called the center. adius the distance from the center to a point on the circle. iameter the distance across a circle, through the center. The diameter is twice the radius. hord a segment whose endpoints are points on the circle. Secant a line that intersects a circle in two points. Tangent a line in the plane of a circle that intersects the circle in exactly one point. oncentric circles coplanar circles that have a common center. Some real life examples are targets and Target! oint of tangency the point at which the tangent line intersects the circle. Syllabus Objective: The student will explore relationships among circles and external lines or rays. Example: Find the length of chord. O 6 0 N x O = O = 6. raw ON. ON bisects ; ON bisects O. Using the properties of triangles, ON = 3 and N = 3 3, so ( ) = 3 3 = 6 3. Unit 0 ircles age of 0
2 Theorem: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem: In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. Example: and are tangent to O at and, respectively. O If = and = 8, then O =?. Since passes through the center, it is perpendicular to the tangent line. The diameter is, so the radius 8 6, ( O ) the ythagorean Theorem to find O. + =O O = 0. is 6. Use Theorem: If two segments from the same exterior point are tangent to a circle, then they are congruent. {Think of an ice cream cone, the two sides should be equal where they meet the ice cream.} Example: a) Find the perimeter of the polygon. I H.5 G J F E 3.5 = J = 3, = =.5, E = EF =, GF = GH =, IH = IJ =.5. dd them all together, = 0 units. Syllabus Objective: The student will solve problems involving arcs, chords, and radii of a circle. entral angle an angle whose vertex is the center of a circle. Minor arc part of a circle that measures less than 80. Major arc part of a circle that measures between 80 and 360. Unit 0 ircles age of 0
3 Semicircle an arc whose endpoints are the endpoints of a diameter of the circle. semicircle measures exactly 80. rc ddition ostulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Theorem: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Examples: a) In E, find the measure of the angle or the arc named. 80 E Solutions: m = m E = 80. m = m = 70. m = m + m = = 50. m = 360 m = = 90. b) In with diameter S, find the measure of the angle or the arc named. S T 45 Q 60 Solutions: Solutions: Q 60 SQ 40 ST 45 T 35 SQ 80 T 35 SQ 0 ST 35 SQ 0 TSQ 65 S 80 Theorem: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Theorem: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Unit 0 ircles age 3 of 0
4 Examples: a) In, SQ = and T = 8. Find. S T Q ST = TQ = SQ = ( ) = 6, ( S) = ( ST ) + ( T ) 6 8, ΔTS is a right triangle, so S = 0. = +, ( S ) is a diameter, so = = 0. b) mgmj = 00, mgk =?. G K H Q M J Since KM is perpendicular to GJ, it also bisects GJ and GJ = 60, mgj = 60, mgk = mgj = ( 60 ) = 80. c) and are tangent to O at and, respectively., if m = 50, then m O =? Δ must measure = 30 to be shared evenly by O and. m = 65, m O = = 5. m O = 5. Theorem: In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Unit 0 ircles age 4 of 0
5 Example: In O, FL = 3, GO = 5, O = 4. Find HJ. G L O F J H L is the midpoint of FG, so LG = 3. ( OL) ( OL) In rt. Δ OLG, 5 = 3 +. Then 5 9 = 6 =, OL = 4. Since O = 4, OL = O an d FG HJ, HJ = FG = FL = 6. Syllabus Objective: 0.  The student will solve problems involving angles, arcs, or sectors of circles. One easy way for students to remember the angle measures relationships: have them imagine a rubber band being held down on two points of the circle. s the rubber band is stretched to the center of the circle it will form an angle, which is equal to the arc measure. Stretched some more and it will be thinner (onehalf the sum of the arcs). Stretched even further, to the edge of the circle and it will be thinner (onehalf the arc). nd stretched further again, even thinner still (onehalf the difference of the arcs). Inscribed angle an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted arc the arc that lies in the interior of an inscribed angle and has endpoints on the angle. Inscribed polygon drawn inside. polygon is inscribed in a circle if all its vertices lie on the circle. ircumscribed drawn around. In the above definition, the circle is circumscribed about the polygon. Measure of an Inscribed ngle Theorem: If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. { angle = arc } Unit 0 ircles age 5 of 0
6 Example: Find the values of x, y, and z. 90 x y x = ( 80 ) = 40, y = 55, y = 0, z = = 80. ( 0) z Theorem: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Example: If m = 0, then m =? and m =?. Since and both intercept. m = m = m = 60. Theorem: If a right triangle is inscribed is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. { angle = arc } Example: If E is a diameter, then E is a? and m 3 =?. 3 E E is a semicir cle, so m 3 = 90. Unit 0 ircles age 6 of 0
7 Theorem: quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Example: If m 4 = 75, then m 5 =?. 4 Since opposite angles of an inscribed quadrilateral are supplementary, m 5 = = Syllabus Objective: The student will solve problems involving secant segments and tangent segments for a circle. T angent segment a segment that is tangent to a circle at an endpoint. tangent segment Secant segment a segment that intersects a circle in two points, with one point as an endpoint of the segment. secant segment E xternal segment the part of the secant segment that is not inside the circle. external segment Theorem: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. { angle = arc } Unit 0 ircles age 7 of 0
8 Example: If X is tangent to and m XZY = 60, find mxy and m XY. Z X mxy = 360 mxzy = = 00. m XY = mxy = ( 00) = 50. Y Theorem: If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of th e measures of the arcs intercepted by the angle and its vertical angle. { angle = ( arc + arc )} Exa mples: a) If mq = 45 and ms = 75, find m. Q m = ( ms + mq ) = ( ) = 60. S b) If m = 55 and ms = 80, find mq. m = ( ms + mq) 55 = ( 80 + mq) 0 = 80+ m Q 30 = mq. Theorem: If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the m easure of the angle formed is one half the difference of the measure of the intercepted arcs. { angle = ( larger arc smaller arc )} Unit 0 ircles age 8 of 0
9 Examples: a) If m = 00 and me = 40, find m. m = ( m me ) E = ( 00 40) = 30 SecantSecant b) c) If m W = 65 and mxz = 70, find mxvy. Y m W = ( mxvy mxz ) Z 65 = ( mxvy 70) W 30 = mx VY = mx VY. X V SecantTangent If mqs = 40, find mqs and m. S mqs + mqs = 360, 40 + mqs = 360, mqs = 0. m = ( mqs mqs ), Q = ( 40 0 ) = 60. TangentTangent Theorem: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the products of the lengths of the segments of the other chord. Unit 0 ircles age 9 of 0
10 Examples: a) If = 4, = 6, and = 8, find. i = i ( ) = ( ) =. b) If = 4, = 9, and = 5, find. Let = x ; then = 5 x. i = i x ( ) = x ( x) = 5x x 5x + 36 = 0 ( x )( x ) 3 = 0 x = or x = 3 = or = 3. Theorem: If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. {(entire length)(outside length)=(entire length)(outside length)} For both of the external segment theorems, students must remember; (entire length)(part outside) = (entire length)(part outside). This works for the tangent segments as well because the segment length is also the external segment length (tangent squared). Unit 0 ircles age 0 of 0
11 Examples: F E G a) If F = 7, EF = 3, and GF = 6, then HF =?. F ief = HF igf, ( 7)( 3) = ( HF )( 6) HF = 8.5. b) If HG = 8, GF = 7, and F =, then EF =?. F ief = HF igf, ( )( EF ) = ( + )( ) 8 7 7, EF = 5., Theorem: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment. {(entire length)(outside length)=(entire length)(outside length)} H Examples: Q S T a) If T = 5 and S = 5, find Q. {(entire length)(outside length)=(entire length)(outside length)} T is = Q iq T is = Q ( )( ) = ( Q ) 5 5, 5 5 = Q. or, b) If Q = 6 and ST = 9, find S. Let S = x ; then T = x + 9, T is = ( Q ) ( x + )( ) =, 9 9 6, x + 9x 36 = 0, ( x + )( x 3) = 0, x = or x = 3, so S = 3. Unit 0 ircles age of 0
12 Syllabus Objective: The student will solve problems involving properties circles using algebraic techniques. Examples: a) Find the value of x. 4 3x+7 5x9 4 Since the arcs are equal (congruent), the chords are congruent. 3x + 7 = 5x 9, 6 = x, x = 8. b) Find m and m. (x8) (3x+) Since is inscribed in a circle, opposite angles are supplementary. m + m = 80, 3x + + x 8 = 80, 44x + 4 = 80, 44x = 76, x = 4. ( ) So, m = 34 + = 04 and ( ) m = 4 8 = 76. c) Find x and y. ssume that segments that appear to be tangent are tangent. ound to the nearest tenth if necessary. Two tangent segments are equal so, 5x8 y 4 5x 8 = 7 3 x, 8x = 80, x = 0. adius is perpendicular to tangent line so, y + 39 = 4, y = 60 = x 39 Unit 0 ircles age of 0
13 d) Find x. J x+ x x+0 K x+8 M L JK ikl = K ikm ( )( ) ( )( x + 0 x = x + x = x x x x 0x = 9x + 8 x = 8. ) Syllabus Objective: The student will perform constructions involving special relationships within circles. {May require supplemental material} Example: onstruct a tangent to a given circle from a point on the circle. O t Given: oint on O. Step : raw O. Step : onstruct a perpendicular from a point ON the line, named t. Line t is tangent to O at. **Eggs** Give students the construction steps and see if they can recreate the shape. The beautifully smooth egg shown below is composed of circular segments that fit together continuously, so that there is no sudden change in the direction of the shell, where any two segments touch. The roblem : econstruct the egg yourself, using only a compass and a straightedge. Unit 0 ircles age 3 of 0
14 Notes : G F N E The order of construction would be:. raw two congruent circles, the second having its center on the circumference of the first.. Join their centers. ( ) 3. Join their points of intersection. (E the perpendicular bisector of ) 4. Mark the intersection of and E. (the midpoint of as N) 5. Mark off point on the line segment N, such that N = N = N. 6. Lengthen and to cut the inside of the original circles at F and G respectively. (and form Δ ) 7. Use compasses centered at, radius F, to draw minor circular FG. 8. Use compasses centered at N, radius N, to draw semicircular. 9. Go over FG, G, and F to emphasize the required Egg shape. Syllabus Objective: The student will graph a circle and determine its equation. Standard equation of a circle a circle with radius r and center (h, k) has this standard equation: ( ) ( ) x h + y k = r Examples: Write the equation of the circle. a) enter at (, 8), radius 7. ( x h) ( y k) ( x ) y ( ) ( x ) ( y ) + = r, ( ) ( ) + 8 = 7, = 7. Unit 0 ircles age 4 of 0
15 b) enter at (, 4), passes through ( 6, 7 ) ( ). Use distance formula to find radius: center to point on circle. r = + ( 6 ) (7 4), r = + ( 4) (3), r = 6+ 9 = 5 = 5. ( x h) + ( y k) = r, ( x ( )) + ( y ) = ( ) ( x + ) + ( y 4) = , Unit 0 ircles age 5 of 0
16 This unit is designed to follow the Nevada State Standards for Geometry, S syllabus and benchmark calendar. It loosely correlates to hapter 0 of Mcougal Littell Geometry 004, sections The following questions were taken from the nd semester common assessment practice and operational exams for and would apply to this unit. Multiple hoice # ractice Exam (0809) Operational Exam (0809) 3. Which accurately describes a tangent?. segment whose endpoints are on the circle.. line that intersects a circle in two points and passes through the center of the circle.. segment having an endpoint on the circle and an endpoint at the center of the circle.. line that intersects a circle at exactly one point. 4. Use the figure below. Which accurately describes a secant?. line that intersects a circle at two points.. segment whose endpoints are on the circle.. segment having an endpoint on the circle and an endpoint at the center of the circle.. line that intersects a circle at exactly one point. Use the figure below. E G E G W ich of the following represent a secant? h. G. E.. 5. In circle S below, Q 3 S Which represents a chord?.... G Use circle O below. Q 86 O T T The m QT = 3, what is the measure of QT? Since m Q= 86, what is the measure of T? Unit 0 ircles age 6 of 0
17 6. In circle J below, L 56 Use circle J below. L J ( 3x 6) K J 66 ( x 4) What is the value of x? In K, =, mxy ( 7x 9) = 3( x + 5) mwz, and m XLY = 48. K What is the value of x? In, ms ( x 9) and m QS = 3. = +, mtu ( x 7) = +, W X S K L Q Z Y T U What is the value of x? In the figure below, m = 75 and m= 35, Q What is the value of x? In the figure below, mmn = 50, M J L m H = 3 and H K What is m? N What is the mjk? Unit 0 ircles age 7 of 0
18 9. Two tangents are drawn from point to circle H. N Two tangents are drawn from point to circle. H What conclusion is guaranteed by this diagram?. mn m N =. ΔHN is a right triangle.. HN is a rhombus.. HN is a kite. 0. ll of the segments shown in the figure below are tangents to N. T What conclusion is guaranteed by this diagram?. m m =. Δ is a right triangle. =. Δ is an isosceles triangle ll of the segments shown in the figure below are tangents to K. G 5 cm 6 cm N 0 cm K 8 cm W 4 cm V Given the measures in the figure above, what is the perimeter of quadrilateral?. 3 cm. 40 cm. 46 cm. 5 cm U 3 cm 3cm I E Given the measures in the figure above, what is the perimeter of quadrilateral GHI?. 8 cm. 6 cm. 3 cm. 36 cm H cm Unit 0 ircles age 8 of 0
19 . In K, NK = 3x + 4, KW = 5x 8, S = 5x 4, and KN KW. N In, FE, H = 4x +, and FE = 6x + 0. E G K F W H What is N? K is the diameter of O, and mjk ( 9( x ) 6) = +. O S =, mj ( 9x) What is the value of x? S is the diameter of M, m MT = ( x+ 5), m UM= ( 3x+ 5), and m SMT = 4x 0. S ( ) J K L What is the value of x? U M What is the value of x? T Unit 0 ircles age 9 of 0
20 48. In circle below, is tangent to at, and is tangent to at. What is the length of? In the figure below, is tangent to at and is tangent to at. ( x + 5) x 6 3( x 7) 6 5x 0 In the figure below, is tangent to at and is tangent to at. x Find the value of x In circle below, MN is tangent to N and M is tangent to at. M 3x N 7 at What is the value of x? In the figure below, is tangent to the circle at and S is a secant. S What is the value of x?. 48 cm. 84 cm. 4 3 cm. cm 8 cm V x 6 cm What is the length of M? In the figure below, is tangent to the circle at and is a secant. 6x What is the value of x?. cm. 5 cm. 5 cm. 5 cm x 0 cm 5 cm Unit 0 ircles age 0 of 0
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