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1 CONGRUENT CHORDS IN A CIRCLE PROBLEM 1aPROBLEM 1b PROBLEM 2aPROBLEM 2b PROBLEM 3aPROBLEM 3b PROBLEM 5aPROBLEM 6b PROBLEM 4 EQUIDISTAN CHORDS FROM CENTER OF CIRCLE Standard 21 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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2 Standard 21: Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. Los estudiantes prueban y resuelven problemas relacionados con cuerdas, secantes, tangentes, ángulos inscritos y polígonos inscritos y circunscritos a círculos. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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3 M L is the Arc of chord If then In congruent circles as in a circle, two arcs are congruent if and only if their corresponding chords result to be congruent. En un círculo o en circulos congruentes, dos arcos son congruentes si y solo si sus cuerdas correspondientes son congruentes. R LM P S Q A C B PS AB PS AB Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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4 P B A In. P then BD DA and BC CA If chord BA In any circle, if the diameter is perpendicular to a chord, it results that it bisects the chord and its arc. En un círculo, si un diámetro es perpendicular a una cuerda, entonces biseca la cuerda y el arco. E D C isto diameter CE Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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5 Find the value of X 20 x 14 20 7 7 20 = 2 7 + 2 X 2 400 = 49 + X 2 -49 X = 351 2 2 X = 18.7 X= 18.7 and X=-18.7 X=? Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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6 Find the value of X 16 x 18 16 9 9 16 = 2 9 + 2 X 2 256 = 81 + X 2 -81 X = 175 2 2 X = 13.2 X= 13.2 and X=-13.2 X=? Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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7 Suppose the diameter of a circle is 46 feet long and a chord is 20 feet long. Find the distance from the chord to the center of the circle. 46 20 X 10 23 X 23 = 2 10 + 2 X 2 529 = 100 + X 2 -100 X = 429 2 2 X = 20.7 X= 20.7 and X=-20.7 Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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8 Suppose the diameter of a circle is 68 feet long and a chord is 32 feet long. Find the distance from the chord to the center of the circle. 68 32 X 16 34 X 34 = 2 16 + 2 X 2 1156 = 256 + X 2 -256 X = 900 2 2 X = 30 X= 30 and X=-30 Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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9 Suppose a chord is 6 inches long and is 5 inches from the center. Find the length of the radius. 6 5 r 5 r 3 3 r = 2 5 + 2 3 2 r = 34 2 2 r = 5.8 r= 5.8 and r=-5.8 r = 25 + 9 2 Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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10 Suppose a chord is 8 inches long and is 6 inches from the center. Find the length of the radius. 8 6 r 6 r 4 4 r = 2 6 + 2 4 2 r = 52 2 2 r = 7.2 r= 7.2 and r=-7.2 r = 36 + 16 2 Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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11 Complete the following proof C B D A T E Given: Circle T CD EB Prove: CA DA CB DB Statements Reasons a. Draw radiiand Through any 2 pts. there is 1 line. b. Given c. TC TD All radii are d. Reflexive property e. HL CA DA f. CPCTC g. CB DB Definition of arcs TC TD Circle T, CD EB TA CAT DAT CTA DTA Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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12 Complete the following proof C B D A T E Given: Circle T CD EB Prove: CA DA CB DB Statements Reasons a. Draw radiiand Through any 2 pts. there is 1 line. b. Given c. TC TD All radii are d. Reflexive property e. HL CA DA f. CPCTC g. CB DB Definition of arcs TC TD Circle T, CD EB TA CAT DAT CTA DTA Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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13 P Q A B D C R If PR QR then AB DC In a circle or in congruent circles, two chord are congruent if and only if they are equidistant from the center. En un círculo o en círculos congruentes, dos cuerdas son congruentes si y solo si ellas son equidistantes del centro. Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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14 K S L R A If KLm find ARS m =110° and Since chords and RS KL are equally distanced from the center of the circle KL RS then KLm RSm = m = 110° Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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15 K S L R A If KLm find ARS m =110° and Since chords and RS KL are equally distanced from the center of the circle KL RS then KLm RSm = m = 110° RAS m 110° RSm = RAS m = 110° RASis isosceles because RA and SA are radii and all radii in a circle are congruent. and then = m ARS ASR m RAS m ASR m ARS m + + = 180° ARS m 2 + 110° = 180° ARS m 2 + 110° = 180° -110° ARS m = 70° 2 2 ARS m=35° Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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16 Q M P L A If QPm find ALM m =120° and Since chords and LM QP are equally distanced from the center of the circle QP LM then QPm LMm = m = 120° Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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17 Q M P L A If QPm find AML m =120° and Since chords and LM QP are equally distanced from the center of the circle QP LM then QPm LMm = m = 120° LAM m 120° LM m = LAM m = 120° LAMis isosceles because LA and MA are radii and all radii in a circle are congruent. and then = m ALM AML m LAM m AML m ALM m + + = 180° AML m 2 + 120° = 180° AML m 2 + 120° = 180° -120° AML m = 60° 2 2 AML m=30° Standard 21 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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