Chapter 6 Test Review
Chapter 6 Test - Review Chapter 6 Test - Review x 115° X = _____ The arc measure of a minor arc is the same as the measure of the central angle, the angle with its vertex at the center of the circle, and sides passing through the endpoints of the arc. 115° S R P v 115° Q r Line r is tangent to both circles. 65° v = ____. Tangent Conjecture A tangent to a circle is perpendicular to the radius drawn to the point of tangency. The angles at the point of tangencies S and R are 90°. Therefore, the sum of angle v + 115° = 180°, making v = 65° Chapter 6 Test - Review Ray c is tangent to circle T. Find X c S 112° 68° T Linear Pair Tangent Conjecture A tangent to a circle is perpendicular to the radius drawn to the point of tangency. X R Solution 1: Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles The angle at the point of tangency S is equal to 90°. Therefore: 112° = X + 90° 22° = X Solution 2: The angle at the point of tangency S is equal to 90°. Linear Pair: 112°+ 68° = 180° Therefore: 180° = X + 90° + 68° 22° = X Chapter 6 Test - Review A B Find the value of X C Chord Central Angles Conjecture If two chords in a circle are congruent, then they determine two central angles that are congruent. 78° X O D Another solution is that you have two congruent isosceles triangles. The radii are congruent and the chords are congruent. So by CPCTC, X = 78° Find the value of X 290° Chord Arcs Conjecture If two chords in a circle are congruent, then their intercepted arcs are congruent The arcs have to add up to 360°. So, the missing arc is 35° making the two arcs congruent. O X Therefore, X = 78° 6 in. 35° Therefore the chords are congruent. So , X = 6 in. Chapter 6 Test - Review Find the value of x and y O x y 68° x The arc measure of a minor arc is the same as the measure of the central angle, the angle with its vertex at the center of the circle, and sides passing through the endpoints of the arc. Therefore, if the arc measure is 68°, y = 68°. Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles Since the radii make the triangle an isosceles triangle 68° = X + X 68° = 2X 34° = X Chapter 6 Test - Review If the mRST = 40°, what is the mRQT ? R 80° 80° S 40° Q T Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. If the mRST = 40°, then measure of the intercepted arc RT = 80° The arc measure of a minor arc is the same as the measure of the central angle. Therefore, if the arc measure is 80°, mRQT = 80°. Chapter 6 Test - Review If the mRST = 47°, what is the mRST ? R 94° S 47° Q T Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. If the mRST = 47°, then measure of the intercepted arc RT = 94° The arc measure of a minor arc is the same as the measure of the central angle. Therefore, if the arc measure is 94°, mRQT = 94° and the reflex measure of the central angle = 266° making mRST = 266° Chapter 6 Test - Review Find x, y and z. z K L 88° 95° 102° Q x J y M Cyclic Quadrilateral Conjecture The opposite angles of a cyclic quadrilateral are supplementary. Therefore, x = 180° - 88° x = 92° And y = 180° - 95° y = 85° Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. Therefore, if x = 92°, the mKLM = 184° And z = 184° - 102° z = 82° Chapter 6 Test - Review KP and RA are parallel. Find mKAR K Parallel Lines Intercepted Arcs Conjecture Parallel lines intercept congruent arcs on a circle. Q P 84° A R Therefore, if mAP = 84°, then mKR = 84° Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. Therefore, the mKAR = 42° Chapter 6 Test - Review If C = 14 cm, find d. C= d 14 cm = d 14 cm = d If d = 12 m, find C. Leave the answer in terms of . C= d C= (12m) C= 12 m What is the approximate circumference of circle Q? Round to the nearest whole unit. Q C= 2r C= 2(45) in. C= 90 in. C 90(3.14) in. C 282.7 in. C 283 in. Chapter 6 Test - Review The circumference of a circle is 35 meters. What is the approximate radius of the circle? Round to nearest 0.01 unit. C= 2r 35 m = 2r 35 m 2(3.14)r 35 m 6.28r 𝟑𝟓 𝐦 r 𝟔.𝟐𝟖 5.57 m r What fraction of the circle is JK. J The arc measure of a minor arc is the same as the measure of the central angle. Q 20 cm K Therefore mJK = 90°. and, 𝟗𝟎° 𝟑𝟔𝟎° = 𝟗 𝟑𝟔 = 𝟏 𝟒 Chapter 6 Test - Review (𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑐) The length of CPU is ____. Arc Length = 360 C 𝟗𝟎° 𝟗 𝟏 The ratio of CU = = = 90° 𝟑𝟔𝟎° Q 45 cm U 𝟑𝟔 So the ratio of CPU is: 𝟒 𝟑 𝟒 𝟑 𝟒 𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝑷𝑼 C P 𝟑 𝟒 𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝑪𝑷𝑼 (283) cm 𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝐂𝐏𝐔 212.25 cm C C= 2r C= 2(45 cm) C= 90 cm C 90(3.14) C 282.7 cm C 283 cm Chapter 6 Test - Review The distance around the equator of Mars is about 21,344 km. What is the equatorial diameter of Mars? (Round to the nearest whole unit) C= d 21,344 km = d 21,344 km = d 21,344 km d 3.14 6797.45 km d 6797 km d Chapter 6 Test - Review Given the mX=110° and mY=110° , find the mZ X W Y Cyclic Quadrilateral Conjecture The opposite angles of a cyclic quadrilateral are supplementary. Therefore, Z = 180° - 110° Z = 70° Z Find the measure of the unknown angles. Cyclic Quadrilateral Conjecture N O The opposite angles of a cyclic quadrilateral 86° b a 85° are supplementary. c M d = 86°, b = 94° c = 85°, a = 95° d P Note that since a is supplementary with 85° then c = 85°. And since b is supplementary with 86° then d = 86°. Chapter 6 Test - Review Amanda ran 9 times around the circular track that has a radius of 57 meters. What is the approximate distance that she ran? (Round to the nearest meter) Run 9 times around the track. 57 m Find the Circumference: C= 2r C = 2(57 m) C = 114 m C 114(3.14) m C 358.1 m C 358 m Find the Distance: d = (C)(number of times around track) d (358)(9) m d 3,222 m Does the following description represent the arc length or arc measure? The angle formed by two different radii in a circle. Arc Measure Central ° Angle Chapter 6 Test - Review Does 95° represent arc measure? Explain why or why not. The measure of an arc is measured in degrees, while the measure of the arc length is measured in distance. So, yes, 95° represents an arc measure. 𝟏 𝟑 Which of the following show of a circle. J 𝟗𝟎° 𝟑𝟔𝟎° L J = 120° 𝟏𝟐𝟎° 𝟑𝟔𝟎° 𝟏 𝟒 K = 𝟏 𝟑 K L 𝟏𝟖𝟎° 𝟑𝟔𝟎° = J 𝟏 𝟐 K J 160° M 𝟏𝟔𝟎° 𝟑𝟔𝟎° L L M = 𝟒 𝟗 K Chapter 6 Test - Review What is the arc length if the radius is 15 meters? Length of QD _____. Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. Q 8𝟎° Arc Length = D 40° (𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄) 𝟑𝟔𝟎 (𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄) 𝟑𝟔𝟎 C = 2r C=2 (15 m) C=30 m 𝟖𝟎 𝟐 = 𝟑𝟔𝟎 𝟗 𝟐 Arc Length = 𝟗 C C 𝟐 (30 m) 𝟗 𝟐 Arc Length = (10 m) 𝟑 𝟐𝟎 Arc Length = m 𝟑 Arc Length = Chapter 6 Test - Review Elizabeth watched a bug crawl through an arc of 18° along the rim of half a melon. If the radius of the melon was 8 inches, how far did the bug crawl? Arc Length = (𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄) 𝟑𝟔𝟎 (𝒎𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒂𝒓𝒄) 𝟑𝟔𝟎 8 in. 18° 𝟏𝟖 𝟑𝟔𝟎 𝟗 = 𝟏𝟖𝟎 = C = 2r C=2 (8 in) C=16 in C 16(3.14) in C 50.27 in 𝟏 𝟐𝟎 Arc Length C 𝟏 𝟐𝟎 C 𝟏 Arc Length (50.24 in) 𝟐𝟎 Arc Length 2.512 in Arc Length 2.512 in Chapter 6 Test - Review In circle Q, PA and PB are tangents. Find mP A P Q 100° 260° The measure of an angle is the smallest amount of rotation about the vertex from one ray to the other, measured in degrees. According to this definition, the measure of an angle can be any value between 0° and 180°. The largest amount of rotation less than 360° between the two rays is called the reflex measure of an angle. 260° is the reflex measure of Q. B Therefore m Q = 100° Tangent Conjecture A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Since A and B are right angles, then Q and P are supplementary. mQ + mP = 180° 100°+ mP = 180° mP = 180° - 100° mP = 80° Chapter 6 Test - Review CD = EF QM = 7 cm QN = _____ D Chord Distance to Center Conjecture Two congruent chords in a circle are equidistant from the center of the circle M F Q Therefore, QM = QN. N C E So, QN = 7 cm Chapter 6 Test - Review If the mRST = 264°, what is the m RST ? R If the mRST = 264°, then m RT = 96° 96° S 264° 48° Q T The arc measure of a minor arc is the same as the measure of the central angle. Therefore, if the arc measure is 96°, mRQT = 96° Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. Therefore mRST = 48°, Chapter 6 Test - Review
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