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Transcript
Eleanor Roosevelt High School Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin A circle is the set of all points in a plane that are equidistant from a fixed point of the plane called the center of the circle Circles are named by their center (e.g., Circle C) Symbol: O C ERHS Math Geometry Circle Mr. Chin-Sung Lin It is the center of the circle and the distance from this point to any other point on the circumference is the same C Circle Center ERHS Math Geometry Mr. Chin-Sung Lin A radius is the line segment connecting (sometimes referred to as the “distance between”) the center and the circle itself C r Center Radius ERHS Math Geometry Circle A Mr. Chin-Sung Lin A circumference is the distance around a circle It is also the perimeter of the circle, and is equal to 2 times the length of radius (2 r) Circumference C r Center Radius ERHS Math Geometry Circle A Mr. Chin-Sung Lin A chord is a line segment with endpoints on the circle Circle C B ERHS Math Geometry A Chord Mr. Chin-Sung Lin A diameter of a circle is a chord that has the center of the circle as one of its points B ERHS Math Geometry C Diameter Circle A Mr. Chin-Sung Lin An arc is a part of the circumference of a circle (e.g., arc AB) A C Arc Circle B ERHS Math Geometry Mr. Chin-Sung Lin A central angle is an angle in a circle with vertex at the center of the circle (e.g., ACB) A C Arc Circle B ERHS Math Geometry Mr. Chin-Sung Lin Given two points on a circle, the major arc is the longest arc linking them (e.g., arc ADB, m ACB > 180) Major Arc A C D Circle B ERHS Math Geometry Mr. Chin-Sung Lin Given two points on a circle, the minor arc is the shortest arc linking them (e.g., arc AB, m ACB < 180) A Minor Arc C Circle B ERHS Math Geometry Mr. Chin-Sung Lin Half a circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle (e.g., arc ADB, m ACB = 180) Semicircle D B ERHS Math Geometry C Circle A Mr. Chin-Sung Lin Adjacent arcs are non-overlapping arcs with the same radius and center, sharing a common endpoint (e.g., arc AB and AD) D Adjacent Arcs A Circle C B ERHS Math Geometry Mr. Chin-Sung Lin Intercepted Arc is the part of a circle that lies between two lines that intersect it (e.g., arc AB and XY) A X Intercepted Arcs Circle C Y ERHS Math Geometry B Mr. Chin-Sung Lin An arc length is the distance along the curved line making up the arc A Circle Arc Length C B ERHS Math Geometry Mr. Chin-Sung Lin The degree measure of an arc is equal to the measure of the central angle that intercepts the arc (e.g., m AB = m ACB) A Circle C B ERHS Math Geometry Measure of Central Angle = Measure of Intercepted Arc Mr. Chin-Sung Lin The measure of minor arc is the degree measure of central angle of the intercepted arc (e.g., m AB = m ACB) A Circle C Degree Measure of a Minor Arc B ERHS Math Geometry Mr. Chin-Sung Lin The measure of major arc is 360 minus the degree measure of the minor arc (e.g., m ADB = 360 – m ACB) D Degree Measure of a Major Arc A Circle C B ERHS Math Geometry Mr. Chin-Sung Lin Congruent circles are circles that have congruent radii (e.g., O ≅ O’) Congruent B Circles O’ O Circle ERHS Math Geometry A Circle Mr. Chin-Sung Lin Congruent arcs are arcs that have the same degree measure and are in the same circle or in congruent circles (e.g., AB ≅ CD ≅ XY) X O’ O Y Circle ERHS Math Geometry A C D Congruent Arcs B Circle Mr. Chin-Sung Lin Concentric Circles are two circles in the same plane with the same center but different radii A O X ERHS Math Geometry Concentric Circles Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin In the same or congruent circles all radii are congruent If C O, r, s and t are radii, then r = s = t C r O s t Congruent Radii ERHS Math Geometry Mr. Chin-Sung Lin In the same or in congruent circles, if two central angles are congruent, then the arcs they intercept are congruent If central angles ACB XOY, A then the intercepted arcs AB X XY Congruent Central Angles = Congruent Arcs ERHS Math Geometry C B O Y Mr. Chin-Sung Lin In the same or in congruent circles, if two arcs are congruent, then their central angles are congruent If the arcs AB XY, then their central angles ACB A XOY Congruent Arcs = Congruent Central Angles ERHS Math Geometry C B X O Y Mr. Chin-Sung Lin In the same or in congruent circles, two arcs are congruent if and only if their central angles are congruent The arcs AB XY, A if and only if their central angles ACB XOY Congruent Arcs = Congruent Central Angles ERHS Math Geometry X C B O Y Mr. Chin-Sung Lin If AB and BC are two adjacent arcs of the same circle , then AB + BC = ABC and mAB + mBC = mABC C B Circle O A ERHS Math Geometry Mr. Chin-Sung Lin In the same or in congruent circles, if two central angles are congruent, then the chords are congruent If central angles ACB then the chords AB XY Congruent Central Angles = Congruent Chords ERHS Math Geometry XOY, A C B X O Y Mr. Chin-Sung Lin In the same or in congruent circles, if two chords are congruent, then their central angles are congruent If the chords AB XY, then their central angles ACB Congruent Chords = Congruent Central Angles ERHS Math Geometry XOY A C B X O Y Mr. Chin-Sung Lin In the same or in congruent circles, two chords are congruent if and only if their central angles are congruent The chords AB XY if and only if A their central angles ACB X XOY Congruent Chords = Congruent Central Angles ERHS Math Geometry C B O Y Mr. Chin-Sung Lin In the same or in congruent circles, if two arcs are congruent, then the chords are congruent If arcs AB XY, then the chords AB XY Congruent Arcs = Congruent Chords ERHS Math Geometry A C B X O Y Mr. Chin-Sung Lin In the same or in congruent circles, if two chords are congruent, then their arcs are congruent If the chords AB then their arcs AB XY, XY Congruent Chords = Congruent Arcs ERHS Math Geometry A C B X O Y Mr. Chin-Sung Lin In the same or in congruent circles, two chords are congruent if and only if the arcs are congruent Arcs AB XY if and only if the chords AB XY A Congruent Arcs = Congruent Chords ERHS Math Geometry C B X O Y Mr. Chin-Sung Lin The diameter of a circle divides the circle into two congruent arcs (semicircles) If AB is a diameter of circle C, then APB P B AQB Diameter A C Q ERHS Math Geometry Mr. Chin-Sung Lin Circle C has central angle ACB = 60o, what’s the measure of the arc ADB? A D C B ERHS Math Geometry Mr. Chin-Sung Lin Circle C has central angle ACB = 60o, DCE = 60o, and BCD = 170o, what’s the measure of the arc AD and BE? A D E ERHS Math Geometry C B Mr. Chin-Sung Lin Circle C has diameter BD and EF. If central angle ACF = 90o, DCE = 50o, what’s the measure of the arc DF, AE and BE? E D A C B F ERHS Math Geometry Mr. Chin-Sung Lin The length of the diameter of circle C is 26 cm. The chord AB is 5 cm away from the center C. What is the length of AB? 26 C X A 5 Y B ERHS Math Geometry Mr. Chin-Sung Lin The length of the chord AB of circle C is 10. The circumference of circle C is 20 . What’s the measure of arc AB? A C B ERHS Math Geometry Mr. Chin-Sung Lin If two concentric circles have radii 10 and 6 respectively, what’s the total area of the blue regions? 10 C 6 ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin If a diameter is perpendicular to a chord, then it bisects the chord and its major and minor arcs C Given: Diameter CD AB Prove: 1) CD bisects AB 2) CD bisects AB and ACB Circle O A M B D ERHS Math Geometry Mr. Chin-Sung Lin If a diameter is perpendicular to a chord, then it bisects the chord and its major and minor arcs C Given: Diameter CD AB Prove: 1) CD bisects AB 2) CD bisects AB and ACB Circle O 1 2 A M B D ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin A secant is a segment or line which passes through a circle, intersecting at two points B D ERHS Math Geometry A Secant C Mr. Chin-Sung Lin A tangent is a line in the plane of a circle that intersects the circle in exactly one point (called the point of tangency) D B Point of Tangent C Tangent A ERHS Math Geometry Mr. Chin-Sung Lin There are 360 degrees in a circle or 2 radians in a circle Thus 2 radians equals 360 degrees C ERHS Math Geometry 360o or 2 A Mr. Chin-Sung Lin An inscribed angle is an angle that has its vertex and its sides contained in the chords of the circle (e.g., ADB) A D Inscribed Angle C B ERHS Math Geometry Mr. Chin-Sung Lin An inscribed polygon is a polygon whose vertices are on the circle W Z Inscribed Polygon C X Y ERHS Math Geometry Mr. Chin-Sung Lin Circumscribed polygon is a polygon whose sides are tangent to a circle W Z Circumscribed Polygon C X Y ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin The measure of an inscribed angle is equal to one-half the measure of its intercepted arc C Circle Given: Inscribed angle ACB Prove: mACB = (1/2) m AB O A ERHS Math Geometry B Mr. Chin-Sung Lin The measure of an inscribed angle is equal to one-half the measure of its intercepted arc C Given: Inscribed angle ACB Prove: mACB = (1/2) m AB O 1 A Proof: (Case 1) Inscribed angles where one chord is a diameter ERHS Math Geometry 2 3 B Mr. Chin-Sung Lin The measure of an inscribed angle is equal to one-half C the measure of its intercepted arc Circle 3 4 Given: Inscribed angle ACB Prove: mACB = (1/2) m AB O 1 2 B A Proof: (Case 2) Inscribed angles with the center of the circle in their interior ERHS Math Geometry Mr. Chin-Sung Lin The measure of an inscribed angle is equal to one-half the measure of its intercepted arc Circle Given: Inscribed angle ACB Prove: mACB = (1/2) m AB O D A Proof: (Case 3) Inscribed angles with the center of the circle in their exterior ERHS Math Geometry 1 2 3 C 5 4 6 B Mr. Chin-Sung Lin Given: Inscribed angle ACB Prove: mACB = (1/2) m AB m1 = m3 - m2 m4 = m6 - m5 m3 = 2 m6 m2 = 2 m5 m3 - m2 = 2 (m6 - m5) = 2 m4 m1 = 2 m4 m4 = (1/2) m1 mACB = (1/2) m AB ERHS Math Geometry Circle O D A 5 1 2 3 C 4 6 B Mr. Chin-Sung Lin In the same or in congruent circles, if two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent C Circle D Given: Inscribed angle ACB and ADB Prove: ACB ADB O A ERHS Math Geometry B Mr. Chin-Sung Lin In the same or in congruent circles, if two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent C Circle D Given: Inscribed angle ACB and ADB Prove: ACB ADB O A ERHS Math Geometry B Mr. Chin-Sung Lin An angle inscribed in a semi-circle is a right angle C Given: Inscribed angle ACB and AB is a diameter Prove: mACB = 90o Circle A O B ERHS Math Geometry Mr. Chin-Sung Lin An angle inscribed in a semi-circle is a right angle C Given: Inscribed angle ACB and AB is a diameter Prove: mACB = 90o Circle A O 180o ERHS Math Geometry B Mr. Chin-Sung Lin If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary B Given: ABCD is an inscribed quadrilateral of circle O Prove: mB + mD= 180 Circle C O A D ERHS Math Geometry Mr. Chin-Sung Lin If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary B Given: ABCD is an inscribed quadrilateral of circle O Prove: mB + mD= 180 Circle C O A D ERHS Math Geometry Mr. Chin-Sung Lin In a circle, parallel chords intercept congruent arcs between them Circle Given: AB || CD Prove: AC BD C D O A ERHS Math Geometry B Mr. Chin-Sung Lin In a circle, parallel chords intercept congruent arcs between them Circle Given: AB || CD Prove: AC BD C D O A ERHS Math Geometry B Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin C has an inscribed quadrilateral ABCD where A = 70o and B = 80o. What’s the measures of C and D? B 80o C O A 70o D ERHS Math Geometry Mr. Chin-Sung Lin C has an inscribed quadrilateral ABCD where A = 70o and B = 80o. What’s the measures of C and D? B 80o O A 70o 110o C 100o D ERHS Math Geometry Mr. Chin-Sung Lin C has an inscribed angle ADB = 30o, DB is the diameter. DEA =? E D 30o C B A Mr. Chin-Sung Lin C has an inscribed angle ADB = 30o, DB is the diameter. DEA =? E 60o D 30o C 60o B A Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin At a given point on a circle, there is one and only one tangent to the circle Circle Given P is on the circle O There is only one tangent AP to circle O P O Tangent A ERHS Math Geometry Mr. Chin-Sung Lin If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of contact B Given: AB is a tangent to O P is the point of tangency Prove: AB OP Circle P O A ERHS Math Geometry Mr. Chin-Sung Lin Given: AB is a tangent to O P is the point of tangency E Prove: AB OP (Indirect Proof) 2. Draw a point D on AB, OD AB E is on different side of D Circle D 1. Suppose OP is NOT perpendicular to AB 3. Draw point E on AB, PD = DE and B P O A 4. ODP = ODE = 90° 5. OD = OD (Reflexive) ERHS Math Geometry Mr. Chin-Sung Lin Given: AB is a tangent to O P is the point of tangency E Prove: AB OP (Indirect Proof) P (CPCTC) 8. E is on O (by 7) 9. AB intersects the circle at two Circle D 6. ODP ODE (SAS) 7. OP = OE B O A different points, so AB is not a tangent (contradicts to the given) 10. AB OP (the opposite of the assumption is true) ERHS Math Geometry Mr. Chin-Sung Lin If a line is perpendicular to a radius at its outer endpoint, then it is a tangent to the circle B Circle Given: OP is a radius of O and P AB OP at P Prove: AB is a tangent to O O A ERHS Math Geometry Mr. Chin-Sung Lin Given: OP is a radius of O and AB OP at P Prove: AB is a tangent to O D 1. Let D be any point on AB other than P 2. OP AB (Given) 3. OD > OP (Hypotenuse is longer) 4. D is not on O (Def. of circle) B Circle P O A 5. AB is a tangent to O (Def. of circle) ERHS Math Geometry Mr. Chin-Sung Lin If a line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of contact B Circle P O A ERHS Math Geometry Mr. Chin-Sung Lin A common tangent is a line that is tangent to each of two circles A B O’ O O B O’ A Common Internal Tangent ERHS Math Geometry Common External Tangent Mr. Chin-Sung Lin Two circles can have four, three, two, one, or no common tangents 4 3 ERHS Math Geometry 2 1 0 Mr. Chin-Sung Lin Two circles are said to be tangent to each other if they are tangent to the same line at the same point Tangent Externally ERHS Math Geometry Tangent Internally Mr. Chin-Sung Lin A tangent segment is a segment of a tangent line, one of whose endpoints is the point of tangency PQ and PR are tangent segments of the tangents PQ and PR to circle O from P. Circle R O Tangent P Segments ERHS Math Geometry Q Mr. Chin-Sung Lin If two tangents are drawn to a circle from the same external point, then these tangent segments are congruent P Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AP AQ O A Q ERHS Math Geometry Mr. Chin-Sung Lin If two tangents are drawn to a circle from the same external point, then these tangent segments are congruent P Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AP AQ (HL Postulate) ERHS Math Geometry O A Q Mr. Chin-Sung Lin If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle formed by the tangents P Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AO bisects PAQ O A Q ERHS Math Geometry Mr. Chin-Sung Lin If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle whose vertex is the center of the circle and whose rays are the two radii drawn to the points of tangency. P Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AO bisects POQ O A Q ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin Circles O and O’ with a common internal tangent, AB, tangent to circle O at A and circle O’ at B, and C the intersection of OO’ and AB (a) Prove AC/BC = OC/O’C (b) Prove AC/BC = OA/O’B (c) If AC = 8, AB = 12, and OA = 9 find O’B B C O O’ A ERHS Math Geometry Mr. Chin-Sung Lin Circles O and O’ with a common internal tangent, AB, tangent to circle O at A and circle O’ at B, and C the intersection of OO’ and AB (a) Prove AC/BC = OC/O’C (b) Prove AC/BC = OA/O’B (c) If AC = 8, AB = 12, and OA = 9 find O’B B C O O’ A (c) O’B = 9/2 ERHS Math Geometry Mr. Chin-Sung Lin C has a tangent AB. If AB = 8, and AC = 12. (a) What is exact length of the radius of the circle? (a) Find the length of the radius of the circle to the nearest tenth B 8 A C ERHS Math Geometry D 12 Mr. Chin-Sung Lin C has a tangent AB. If AB = 8, and AC = 12. (a) What is exact length of the radius of the circle? (a) Find the length of the radius of the circle to the nearest tenth B 8 A (a) 4√ 5 (b) 8.9 ERHS Math Geometry C D 12 Mr. Chin-Sung Lin C has a tangent AB and a secant AE. If the diameter of the circle is 10 and AD = 8. AB = ? B A 10 D E ERHS Math Geometry 8 C Mr. Chin-Sung Lin C has a tangent AB and a secant AE. If the diameter of the circle is 10 and AD = 8. AB = ? B 5 E ERHS Math Geometry C A 5 D 8 Mr. Chin-Sung Lin Find the perimeter of the quadrilateral WXYZ Z 4 W D Circumscribed Polygon & Inscribed Circle A C 8 C 5 Y ERHS Math Geometry X B Mr. Chin-Sung Lin Find the perimeter of the quadrilateral WXYZ Z 4 W D A C Perimeter: 34 Circumscribed Polygon & Inscribed Circle 8 C 5 Y ERHS Math Geometry X B Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin Measure an angle formed by A tangent and a chord Two tangents Two secants A tangent and a secant Two chords ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin The measure of an angle formed by a tangent and a chord equals one-half the measure of its intercepted arc Given: CD is a tangent to O, B is the point of tangency, and AB is a chord E Prove: A 1) mABC = (1/2) m AB O 2) mABD = (1/2) m AEB C ERHS Math Geometry B D Mr. Chin-Sung Lin Given: CD is a tangent to O, B is the point of tangency, and AB is a chord Prove: A 1) mABC = (1/2) m AB 1 O 2) mABD = (1/2) m AEB E 2 C ERHS Math Geometry B D Mr. Chin-Sung Lin 1. Draw OA and OB, form 1 and 2 2. OB CD 3. mABC + m2 = 90 A 4. OA = OB 5. m1 = m2 6. m1 + m2 + mAOB = 180 7. 2m2 + mAOB = 180 8. m2 + (1/2) mAOB = 90 9. m2 + (1/2) mAOB = mABC + m2 C 10. (1/2) mAOB = mABC 11. mABC = (1/2) m AB 12. 180 - mABC = (1/2) (360 - m AB) 13. mABD = (1/2) m AEB ERHS Math Geometry E 1 O 2 B D Mr. Chin-Sung Lin If CD is a tangent to O, B is the point of tangency, and ABE is an inscribed triangle what are the measures of ABC, EBD, AB and EAB ? E A 80o C ERHS Math Geometry 70o O B D Mr. Chin-Sung Lin If CD is a tangent to O, B is the point of tangency, and ABE is an inscribed triangle what are the measures of ABC, EBD, AB and EAB ? 60o E A 70o 80o O 160o 140o 80o 70o C ERHS Math Geometry B D Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin The measure of an angle formed by two tangents, by a tangent and a secant, or by two secants equals one-half the difference of the measure of their intercepted arcs B D D B B O A O O A A C C ERHS Math Geometry E C Mr. Chin-Sung Lin Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) B D D B B O O O A A A C E ERHS Math Geometry C E C E Mr. Chin-Sung Lin Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) B 1. Draw BC, form 1 and 2 1 2. AB = AC 3. m1 = m2 = (1/2) m BC O A 4. mA + m1 + m2 = 180 2 5. mA + m BC = 180 E 6. m BC + m BEC = 360 C 7. (1/2) m BC + (1/2) m BEC = 180 8. mA + m BC = (1/2) m BC + (1/2) m BEC 9. mA = (-1/2) m BC + (1/2) m BEC 10. mA = (1/2) (m BEC - m BC ) ERHS Math Geometry Mr. Chin-Sung Lin Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) 1. Draw OB and OC, form 1 and 2 2. OB AB, OC AC 3. mA + m1 + m2 + mBOC = 360 4. mA + 90 + 90 + mBOC = 360 5. mA + mBOC = 180 A 6. mA + m BC = 180 7. m BC + m BEC = 360 8. (1/2) m BC + (1/2) m BEC = 180 9. mA + m BC = (1/2) m BC + (1/2) m BEC 10. mA = (-1/2) m BC + (1/2) m BEC 11. mA = (1/2) (m BEC - m BC ) ERHS Math Geometry B 1 O 2 E C Mr. Chin-Sung Lin Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) 1. Draw BC, form 2 2 2. Extend AC, form 1 3. m2 = (1/2) m BC B O A 4. m1 = (1/2) m BEC 5. mA = m1 - m2 6. mA = (1/2) m BEC - (1/2) m BC 7. mA = (1/2) (m BEC - m BC ) ERHS Math Geometry 1 E C D Mr. Chin-Sung Lin Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC) B D D B B O O O A A A C E ERHS Math Geometry C E C E Mr. Chin-Sung Lin D Given: O with secants AB and AC Prove: mA = (1/2) (m DE - m BC) 1. Draw DC B 1 O A 2. m2 = mA + m1 3. mA = m2 - m1 2 C E 4. m2 = (1/2) m DE, m1 = (1/2) m BC 5. mA = (1/2) (m DE - m BC) ERHS Math Geometry Mr. Chin-Sung Lin Given: O with secants AB and AC Prove: mA = (1/2) (m DE - m BC) D B 3 1. Draw OB, OC, OD and OE 2. OB = OC = OD = OE 4 5 1 A 2 3. m3 = m4, m7 = m8 C 4. m5 = 180 - 2 m3, m9 = 180 - 2 m7 O 9 7 8 E 5. m3 = mBOA + mBAO 6. m7 = mCOA + mCAO 7. m5 + m9 = 180 - 2 m3 + 180 - 2 m7 = 360 - 2(m3 + m7) ERHS Math Geometry Mr. Chin-Sung Lin Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC) 8. m5 + m9 = 360 - 2(mBOA + mBAO + mCOA + mCAO) = 360 - 2(mBOC + mA) 9. m5 + m9 + 2(mBOC + mA) = 360 D 10. m5 + m9 + mBOC + mDOE = 360 B 11. mBOC - mDOE + 2mA = 0 12. 2 mA = mDOE - mBOC A 13. mA = (1/2) (mDOE- mBOC) 14. mA = (1/2) (m DE- m BC) ERHS Math Geometry 3 1 2 C 4 5 O 9 7 8 E Mr. Chin-Sung Lin Given: O with a secant AD and a tangent AC Prove: mA = (1/2) (m DEC - m BC) B D D B B O O O A A A C E ERHS Math Geometry C E C E Mr. Chin-Sung Lin Given: O with a secant AD and a tangent AC D Prove: mA = (1/2) (m DEC - m BC) B 1. Draw BC 2. m2 = mA + m1 2 O A 3. mA = m2 - m1 4. m2 = (1/2) m DEC, m1 = (1/2) m BC 1 E C 5. mA = (1/2) (m DEC - m BC) ERHS Math Geometry Mr. Chin-Sung Lin The measure of an angle formed by two tangents, by a tangent and a secant, or by two secants equals one-half the difference of the measure of their intercepted arcs B D D B B O A O O A A C C ERHS Math Geometry E C Mr. Chin-Sung Lin If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG 40o B D F O A 50o C E G ERHS Math Geometry Mr. Chin-Sung Lin If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG 40o B 65o A 50o D F O 130o 65o C E 25o G ERHS Math Geometry Mr. Chin-Sung Lin If O with a tangent AB, secants AD, GB and GD, calculate m BD, and m BF B D H 20o A O F 40o C E 30o G ERHS Math Geometry Mr. Chin-Sung Lin If O with a tangent AB, secants AD, GB and GD, calculate m BD, and m BF 100o B D H 60o 20o A O F 40o C E 30o G ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin The measure of an angle formed by two chords intersecting inside a circle equals one-half the sum of the measures of its intercepted arcs Given: O with chords AB and CD Prove: mAMC = mBMD = (1/2) (m AC + m BD) A D O M C B ERHS Math Geometry Mr. Chin-Sung Lin Given: O with chords AB and CD Prove: mAMC = mBMD = (1/2) (m AC + m BD) 1. Draw BC 2. 3. 4. 5. 6. 7. A mAMC = mBMD mAMC = m1 + m2 m1 = (1/2) m AC m2 = (1/2) m BD C mAMC = (1/2) m AC + (1/2) m BD mAMC = mBMD = (1/2) (m AC + m BD) ERHS Math Geometry D O 2 M 1 B Mr. Chin-Sung Lin If O with chords AB, CD, AC and BD, calculate m AC and m AD A D O 70o M 60o C 90o ERHS Math Geometry B Mr. Chin-Sung Lin If O with chords AB, CD, AC and BD, calculate m AC and m AD o 130 A D O 80o 70o M 60o C 90o ERHS Math Geometry B Mr. Chin-Sung Lin Measure an angle formed by A tangent and a chord Two tangents Two secants A tangent and a secant Two chords ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin O has a tangent ED and two parallel chords CD and AB. If the inscribed angle DAB = 20o, Find CDE. E D C A ERHS Math Geometry 20o O B Mr. Chin-Sung Lin O has a tangent ED and two parallel chords CD and AB. If the inscribed angle DAB = 20o, Find CDE. E 100o 50o D C 40o 40o A ERHS Math Geometry 20o O B Mr. Chin-Sung Lin C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF = ? mA = ? B 120o A C E D F ERHS Math Geometry Mr. Chin-Sung Lin C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF = ? mA = ? B 120o 30o 60o 60o E C 60o 30o 60o A D 60o F ERHS Math Geometry Mr. Chin-Sung Lin O has two secants CA and CB. If AE = ED and mEAB = 65, find ECB = ? E A C 65o D O B ERHS Math Geometry Mr. Chin-Sung Lin O has two secants CA and CB. If AE = ED and mEAB = 65, find ECB = ? E A C 65o 65o D O 25o 25o B ERHS Math Geometry Mr. Chin-Sung Lin ABCDE is a regular pentagon inscribed in O and BG is a tangent. Find ABG and AFE. C B D O F G A ERHS Math Geometry E Mr. Chin-Sung Lin ABCDE is a regular pentagon inscribed in O and BG is a tangent. Find ABG and AFE. C B D O 36o G 72o A ERHS Math Geometry 72o F 72o o 108 E Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin Measure segments formed by Two chords A secant and a tangent Two secants ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin If two chords intersect within a circle, the product of the measures of the segments of one chord equals the product of the measures of the segments of the other chord A Given: AB and CD are chords of O, two chords intersect at E Prove: AE · BE = CE · DE O C E D B ERHS Math Geometry Mr. Chin-Sung Lin If two chords intersect within a circle, the product of the measures of the segments of one chord equals the product of the measures of the segments of the other chord A Given: AB and CD are chords of O, two chords intersect at E Prove: AE · BE = CE · DE 1 O C 3 2 4 E D B ERHS Math Geometry Mr. Chin-Sung Lin Given: AB and CD are chords of O, two chords intersect at E Prove: AE · BE = CE · DE 1. Connect BC and AD A 1 O C 3 2 2. m1 = m2 (Congruent inscribed angles) 4 3. m3 = m4 (Congruent inscribed angles) B E D 4. CBE ~ ADE (AA similarity) 5. AE/CE = DE/BE (Corresponding sides proportional) 6. AE · BE = CE · DE (Cross product) ERHS Math Geometry Mr. Chin-Sung Lin If O with chords AB and CD, CD = 10, CM = 6, and AM = 8, calculate AB = ? A 8 D O M 6 C ERHS Math Geometry 10 B Mr. Chin-Sung Lin If O with chords AB and CD, CD = 10, CM = 6, and AM = 8, calculate AB = ? AM · BM = CM · DM 8 · BM = 6 · (10 - 6) A 8 BM = 24 / 8 = 3 M AB = 3 + 8 = 11 6 C ERHS Math Geometry D O 4 3 10 B Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin If a tangent and a secant are drawn to a circle from the same external point, then length of the tangent is the mean proportional between the lengths of the secant and its external segment Given: A is an external point to O, AD is a secant and AC A is a tangent of O, Prove: AD · AB = AC2 D B O C ERHS Math Geometry Mr. Chin-Sung Lin If a tangent and a secant are drawn to a circle from the same external point, then length of the tangent is the mean proportional between the lengths of the secant and its external segment Given: A is an external point to O, AD is a secant and AC A is a tangent of O, Prove: AD · AB = AC2 D 2 B O 1 C ERHS Math Geometry Mr. Chin-Sung Lin Given: A is an external point to O, AD is a secant and AC is a tangent of O, Prove: AD · AB = AC2 D 2 B O A 1. Connect BC and CD 1 2. m1 = (1/2) m BC (Tangent-chord angles theorem)C 3. m2 = (1/2) m BC (Inscribed angles theorem) 4. m1 = m2 (Substitution property) 5. mA = mA (Reflexive property) 6. CBA ~ DCA (AA similarity) 7. AB/AC = AC/AD (Corresponding sides proportional) 8. AD · AB = AC2 (Cross product) ERHS Math Geometry Mr. Chin-Sung Lin If O with tangent AC and secant AD, OD = 5 and AB = 6, calculate AC = ? A 6 O B 5 D C ERHS Math Geometry Mr. Chin-Sung Lin If O with tangent AC and secant AD, OD = 5 and AB = 6, calculate AC = ? AB = 6 AD = 16 AC2 = AD · AB AC2 = 16 · 6 AC = 4 √6 ERHS Math Geometry A 6 B 5 O 5 D 4 √6 C Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin If two secants are drawn to a circle from the same external point then the product of the lengths of one secant and its external segment is equal to the product of the lengths of the other secant and its external segment D Given: A is an external point to O, AD and AE are secants A to O Prove: AD · AB = AE · AC B O C E ERHS Math Geometry Mr. Chin-Sung Lin If two secants are drawn to a circle from the same external point then the product of the lengths of one secant and its external segment is equal to the product of the lengths of the other secant and its external segment D Given: A is an external point to O, AD and AE are secants A to O Prove: AD · AB = AE · AC B 1 O C 2 E ERHS Math Geometry Mr. Chin-Sung Lin Given: A is an external point to O, AD and AE are secants to O Prove: AD · AB = AE · AC D B 1 O A C 1. Connect BE and CD 2 E 2. m1 = (1/2) m BC (Inscribed angles theorem) 3. m2 = (1/2) m BC (Inscribed angles theorem) 4. m1 = m2 (Substitution property) 5. mA = mA (Reflexive property) 6. EBA ~ DCA (AA similarity) 7. AD/AE = AC/AB (Corresponding sides proportional) 8. AD · AB = AE · AC (Cross product) ERHS Math Geometry Mr. Chin-Sung Lin If O with secants AC and AE, OC = DE = x, AD = 10 and AB = 8, calculate BC = ? A O B 8 10 D x C x E ERHS Math Geometry Mr. Chin-Sung Lin If O with secants AC and AE, OC = DE = x, AD = 10 and AB = 8, calculate BC = ? AC · AB = AE · AD 8 (2x + 8) = 10 (10 + x) 4 (2x + 8) = 5 (10 + x) 8x + 32 = 50 + 5x 3x = 18 X=6 A O B 8 C 12 10 D 6 E BC = 12 ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin O has a tangent AF and two secants AC and AB. If AD = 3, CD = 9, and AE = 4, find AF = ? BE = ? F C 9 3 D 4 O A E B ERHS Math Geometry Mr. Chin-Sung Lin O has a tangent AF and two secants AC and AB. If AD = 3, CD = 9, and AE = 4, find AF = ? BE = ? F AF2 = AD · AC = 3 · (3 + 9) C 6 9 AF2 = 36 AF = 6 AF2 = AB · AE 3 D 4 O 5 36 = 4 (BE + 4) A E BE = 5 B ERHS Math Geometry Mr. Chin-Sung Lin C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ? A 6 E C 3 M 3 P 2 N D F ERHS Math Geometry Mr. Chin-Sung Lin C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ? A AP · PF = DP · PE 5 6 · 2 = DP · 3 6 DP = 4 C 3 5 M AC = 5 3 CN = (52 - 3.52)1/2 E D P 2 N 4 F ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the origin and a radius with a length of 5. The points (5, 0), (0, 5), (-5, 0) and (0, -5) are points on the circle. What is the equation of the circle? y B (0, 5) P (x, y) y 5 y C (-5, 0) A (5, 0) O x x x D (0, –5) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the origin and a radius with a length of 5. The points (5, 0), (0, 5), (-5, 0) and (0, -5) are points on the circle. What is the equation of the circle? y B (0, 5) x2 + y2 = 52 or x2 + y2 = 25 P (x, y) y 5 y C (-5, 0) A (5, 0) O x x x D (0, –5) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the origin and a radius with a length of r. The points (r, 0), (0, r), (-r, 0) and (0, -r) are points on the circle. What is the equation of the circle? y B (0, r) P (x, y) y r y C (-r, 0) A (r, 0) O x x x D (0, –r) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the origin and a radius with a length of r. The points (r, 0), (0, r), (-r, 0) and (0, -r) are points on the circle. What is the equation of the circle? y B (0, r) P (x, y) y x2 + y2 = r2 r y C (-r, 0) A (r, 0) O x x x D (0, –r) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the (2, 4) and a radius with a length of 5. The points (7, 4), (2, 9), (-3, 4) and (2, -1) are points on the circle. What is the equation of the circle? x=2 B (2, 9) P (x, y) y |y -4| 5 C (-3, 4) A (7, 4) (2, 4) x y=4 |x – 2| D (2, –1) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the (2, 4) and a radius with a length of 5. The points (7, 4), (2, 9), (-3, 4) and (2, -1) are points on the circle. What is the equation of the circle? x=2 P (x, y) y (x – 2)2 + (y – 4)2 = 52 or B (2, 9) |y -4| 5 C (-3, 4) (x – 2)2 + (y – 4)2 = 25 A (7, 4) (2, 4) x y=4 |x – 2| D (2, –1) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the (h, k) and a radius with a length of r. The points (h+r, k), (h, k+r), (h-r, k) and (h, k-r) are points on the circle. What is the equation of the circle? x = h B (h, k+r) P (x, y) y |y -k| r C (h-r, k) A (h+r, k) (h, k) x y=k |x–h| D (h, k–r) ERHS Math Geometry Mr. Chin-Sung Lin A circle with center at the (h, k) and a radius with a length of r. The points (h+r, k), (h, k+r), (h-r, k) and (h, k-r) are points on the circle. What is the equation of the circle? x = h B (h, k+r) P (x, y) y (x – h)2 + (y – k)2 = r2 |y -k| r C (h-r, k) A (h+r, k) (h, k) x y=k |x–h| D (h, k–r) ERHS Math Geometry Mr. Chin-Sung Lin Center-radius equation of a circle with radius r and center (h, k) is P (x, y) (x – h)2 + (y – k)2 = r2 r (h, k) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (x1, y1) and Q (x2, y2). What is the center C (h, k) of the circle? P (x1, y1) r C (h, k) r Q (x2, y2) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (x1, y1) and Q (x2, y2). The center C (h, k) of the circle is the midpoint of the diameter P (x1, y1) r C (h, k) = ( x1 + x 2 2 , y1 + y 2 ) C (h, k) 2 r Q (x2, y2) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (5, 7) and Q (-1, -1). Find the center of the circle, C (h, k) P (5, 7) r C (h, k) r Q (-1, -1) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (5, 7) and Q (-1, -1). Find the center of the circle, C (h, k) P (5, 7) 5 + (-1) 7 + (-1) C (h, k) = ( , ) 2 2 = (2, 3) r C (h, k) r Q (-1, -1) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (x1, y1) and Q (x2, y2). What is the radius (r) of the circle? P (x1, y1) r C (h, k) r Q (x2, y2) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (x1, y1) and Q (x2, y2). The radius (r) of the circle is equal to ½ PQ P (x1, y1) r r = ½ PQ C (h, k) = ½ √ (x2 – x1)2 + (y2 – y1)2 r Q (x2, y2) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (5, 7) and Q (-1, -1). What is the radius (r) of the circle? P (5, 7) r C (h, k) r Q (-1, -1) ERHS Math Geometry Mr. Chin-Sung Lin A circle has a diameter PQ with end-points at P (5, 7) and Q (-1, -1). What is the radius (r) of the circle? P (5, 7) r r = ½ PQ C (h, k) = ½ √ (-1 – 5)2 + (-1 – 7)2 r = ½ (10) =5 ERHS Math Geometry Q (-1, -1) Mr. Chin-Sung Lin (a) Write an equation of a circle with center at (3, -2) and radius of length 7 (b) What are the coordinates of the endpoints of the horizontal diameter? ERHS Math Geometry Mr. Chin-Sung Lin (a) Write an equation of a circle with center at (3, -2) and radius of length 7 (x–3)2 + (y+2)2 = 49 (b) What are the coordinates of the endpoints of the horizontal diameter? (10, -2), (-4, -2) ERHS Math Geometry Mr. Chin-Sung Lin A circle C has a diameter PQ with end-points at P (-2, 9) and Q (4, 1) (a) What is the center (C) of the circle? (b) What is the radius (r) of the circle? (c) What is the equation of the circle? (d) What are the coordinates of the endpoints of the horizontal diameter? (e) What are the coordinates of the endpoints of the vertical diameter? (f) What are the coordinates of two other points on the circle? ERHS Math Geometry Mr. Chin-Sung Lin A circle C has a diameter PQ with end-points at P (-2, 9) and Q (4, 1) (a) What is the center (C) of the circle? (1, 5) (b) What is the radius (r) of the circle? 5 (c) What is the equation of the circle? (x–1)2 + (y–5)2 = 25 (d) What are the coordinates of the endpoints of the horizontal diameter? (-4, 5), (6, 5) (e) What are the coordinates of the endpoints of the vertical diameter? (1, 10), (1, 0) (f) What are the coordinates of two other points on the circle? (4, 9), (-2, 1) ERHS Math Geometry Mr. Chin-Sung Lin Based on the diagram, (a) write an equation of the circle (b) Find the area of the circle ERHS Math Geometry Mr. Chin-Sung Lin Based on the diagram, (a) write an equation of the circle (b) Find the area of the circle (a) (x+4)2 + (y+4)2 = 25 (b) 25π ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin ERHS Math Geometry Mr. Chin-Sung Lin