h 6 Note Sheets L Shortened Key Note Sheets hapter 6: iscovering and roving ircle roperties eview: ircles Vocabulary If you are having problems recalling the vocabulary, look back at your notes for Lesson 1.7 and/or page 69 71 of your book. lso, pay close attention to the geometry notations you need to use to name the parts!! ircles ircle is a set of points in a plane a given distance (radius) from a given point (center). ongruent circles are two or more circles with the same radius measure. oncentric circles are two or more circles with the same center point. F G ypes of Lines [Segments] adius [of a circle] is segment that goes from the center to any point on the circle. the distance from the center to any point on the circle. ll radii of a circle are equal. iameter [of a circle] is chord that goes through the center of a circle. iameter is the longest chord in a circle. the length of the diameter. d = r or ½ d = r. hord is a segment connecting any two points on the circle. Secant line that intersects a circle in two points. angent [to a circle] is a line that intersects a circle in only one point. he point of intersection is called the point of tangency. rcs & ngles rc [of a circle] is formed by two points on a circle and a continuous part of the circle between them. he two points are called endpoints. Semicircle is an arc whose endpoints are the endpoints of the diameter. Minor arc is an arc that is smaller than a semicircle. Major arc is an arc that is larger than a semicircle. Intercepted rc n arc that lies in the interior of an angle with endpoints on the sides of the angle. entral angle n angle whose vertex lies on the center of a circle and whose sides are radii of the circle. Measure of a central angle determines the measure of an intercepted minor arc. radii:,, diameter: chords:,, F secant: G or G tangent: semicircles:, minor arcs:,,,, major arcs:,, F,, central angles with their intercepted arcs: m = m m = m S. Stirling age 1 of 8
h 6 Note Sheets L Shortened Key Lesson 6.1 angent roperties Investigation 1: See Worksheet page 1. angent onjecture tangent to a circle is perpendicular to the radius drawn to the point of tangency. onverse of the angent onjecture line that is perpendicular to a radius at its endpoint on the circle is tangent to the circle. Investigation : See Worksheet page 1. N angent Segments onjecture angent segments to a circle from a point outside the circle are congruent. G Intersecting angents onjecture he measure of an angle formed by two intersecting tangents to a circle is 180 minus the measure of the intercepted arc, x = 180 a. x a a b angent means to intersect in one point. angent ircles ircles that are tangent to the same line at the same point. Internally angent ircles wo tangent circles having centers on the same side of their common tangent. xternally angent ircles wo tangent circles having centers on opposite sides of their common tangent. olygons and ircles he triangle is inscribed in the circle, or the circle is circumscribed about the triangle. ll of the vertices of the polygon are on the circle. he sides are all chords of the circle. he triangle is circumscribed about the circle, or the circle is inscribed in the triangle. ll of the sides of the polygon are tangent to the circle. S. Stirling age of 8
h 6 Note Sheets L Shortened Key xample 1 xample ays r as s are tangents. w =? 180 54 = w angent radius, so m NM = m M = 90 Quad. sum = so 180 110 = x, x = 70 y = 5 cm angents to a circle from a point are congruent. mn = 110 Measure of a central angle equals its intercepted arc. so central angle = w now use the kite formed by = tangent segments and = radii 90 90 54 = w 16 = w mqn = 110 = 50 otal degrees in a circle = xample 3 is tangent to both circle and circle. w = 100 Quadrilater sum = mx = 60 ircle s degree = m = 90 and m = 90 angent radius, w = 90 90 80 = 100 w= m = 100 Measure of central angle = intercepted arc. mx = 100 = 60 S. Stirling age 3 of 8
h 6 Note Sheets L Shortened Key Lesson 6. hord roperties ead top of page 317. hen use the diagrams to define the following. xamples: Non-xamples: entral angle n angle whose vertex lies on the center of a circle and whose sides are radii of the circle. Measure of a central angle = measure of its intercepted arc., and are central angles of circle. Q S Q, QS, S, QS and QS are N central angles of circle. Inscribed angle n angle whose vertex lies on a circle and whose sides are chords of the circle. xamples:, and are inscribed angles. Non-xamples: Q V W X S U Q, SU, and VWX are N inscribed angles. Investigation 3: See Worksheet page 4. hord onjectures If two chords in a circle are congruent, then they determine two central angles that are congruent. their intercepted arcs are congruent. xample 1 If, then. If, then S. Stirling age 4 of 8
h 6 Note Sheets L Shortened Key Lesson 6.3 rcs and ngles ead top of page 34. xample 1 eview (from hapter 1.7): circle measures. semicircle measures 180. Investigation 4: See Worksheet page 6. Inscribed ngle onjecture he measure of an angle inscribed in a circle equals half the measure of its intercepted arc. If m = 9, then m =?. 9 m = 9. he central angle equals its intercepted arc. If m = 9, then m = 46. he inscribed angle equals half its intercepted arc. xample m = 170, find m and m Q. ( ) m = 1 170 = 85 ( ) m Q= 1 170 = 85 n inscribed angle measures ½ the intercepted arc. Q 170 xample 3 m = m = 4, find m and m Q 4 m = 1 i 4 = 1 m Q= 1 i 4 = 1 he inscribed angle equals half its intercepted arc. 4 Q xample 4, and intercept semicircle. Find the measure of each angle. ll are 1 i 180 = 90 m = m = m = 90. S. Stirling age 5 of 8
h 6 Note Sheets L Shortened Key Lesson 6.5 he ircumference/iameter atio ead top of page 335. i is a number, just like 3 is a number. It represents the number you get when you take the circumference of a circle and divide it by the diameter. here is no exact value for pi, so you use the symbol π. You will leave π in the answers to your problems unless they ask for an approximate answer. If they do, use 3.14 or the π key on your calculator. ircumference he perimeter of a circle, which is the distance around the circle. lso, the curved path of the circle itself. ircumference onjecture If is the circumference and d is the diameter of a circle, then there is a number π such that = dπ Since d = r, where r is the radius, then = rπ or πr xample : If a circle has a diameter of 3 meters, what is its circumference? = πd write the formula = π 3 substitute = 3π simplify (pi written last, like a variable) 9.4 m nly an approximate answer! xample : If a circle has a circumference of 0 meters, what is its diameter? xample : If a circle has a circumference of 1π meters, what is its radius? = π r write the formula 1π = π r substitute 1π πr = π π divide to get r. r = 6 m simplify = πd write the formula 0 = πd substitute 0 = d π accurate answer d 6.366 m approximate S. Stirling age 6 of 8
h 6 Note Sheets L Shortened Key Lesson 6.7 rc Length Investigation 8: See Worksheet page 11. Measure of an rc: he measure of an arc equals the measure of its central angle, measured in degrees. rc length: he portion of (or fraction of) the circumference of the circle described by an arc, measured in units of length. rc Length onjecture he length of an arc equals the measure of the arc divided by times the circumference. It is a fraction of the circle! So. Use the formula every time!! rc Length = arc degrees ircumference xample (a): Given central = 90 with radius 1. m = 90 fraction of circle = 90 = 1 4 length of 90 = ( π i 1 ) = 1 ( 4 π ) = 6 π 4 90 90 xample (b): Given central = 180 with diameter 15 cm. m = 180 fraction of circle = 180 = 1 length of 180 = ( 15 π ) = 15 π cm xample (c): Given mf = 10 with radius = 9 ft. fraction of circle = 10 = 1 3 length of 1 F = ( π i 9 ) 3 = 6π ft F 10 S. Stirling age 7 of 8
h 6 Note Sheets L Shortened Key XML : If the radius of a circle is 4 cm and m = 60, what is the length of? XML : he length of is 116π, what is the radius of the circle? m = 10 10 4 = 16π 50.3 cm length of = ( π i ) 10 60 m = 10 = 40 40 116π = ( πi r) 4π 116π = 3 r then 116 π 3 r 1 i 4π = r = 87 10 S. Stirling age 8 of 8