Eleanor Roosevelt High School Chin-Sung Lin

Eleanor Roosevelt High School Chin-Sung Lin
Geometry Chapter 13 Geometry of The Circle Eleanor Roosevelt High School Chin-Sung Lin

Arcs, Angles, and Chords ERHS Math Geometry Mr. Chin-Sung Lin
L25_Circles Arcs, Angles, and Chords ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Circle 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 Circle C ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Center It is the center of the circle and the distance from this point to any other point on the circumference is the same Circle C Center ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Radius A radius is the line segment connecting (sometimes referred to as the “distance between”) the center and the circle itself Circle r C A Center Radius ERHS Math Geometry Mr. Chin-Sung Lin

Circumference A circumference is the distance around a circle
L25_Circles Circumference 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 Circle r C A Center Radius ERHS Math Geometry Mr. Chin-Sung Lin

Chord A chord is a line segment with endpoints on the circle Circle C
L25_Circles Chord A chord is a line segment with endpoints on the circle Circle C A Chord B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Diameter A diameter of a circle is a chord that has the center of the circle as one of its points Circle C B A Diameter ERHS Math Geometry Mr. Chin-Sung Lin

Arc An arc is a part of the circumference of a circle (e.g., arc AB) A
L25_Circles Arc An arc is a part of the circumference of a circle (e.g., arc AB) A Circle C Arc B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Central Angle A central angle is an angle in a circle with vertex at the center of the circle (e.g., ACB) A Circle C Arc B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Major Arc Given two points on a circle, the major arc is the longest arc linking them (e.g., arc ADB, mACB > 180) Major Arc D A Circle C B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Minor Arc Given two points on a circle, the minor arc is the shortest arc linking them (e.g., arc AB, mACB < 180) A Circle C Minor Arc B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Semicircle 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 Circle C B A ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Adjacent Arcs 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

L25_Circles Intercepted Arc 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 B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Arc Length An arc length is the distance along the curved line making up the arc A Arc Length Circle C B ERHS Math Geometry Mr. Chin-Sung Lin

Degree Measure of an Arc
L25_Circles Degree Measure of an Arc 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 Measure of Central Angle = Measure of Intercepted Arc Circle C B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Measure of a Minor Arc The measure of minor arc is the degree measure of central angle of the intercepted arc (e.g., m AB = mACB) A Degree Measure of a Minor Arc Circle C B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Measure of a Major Arc 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

L25_Circles Congruent Circles Congruent circles are circles that have congruent radii (e.g., O ≅ O’) Congruent Circles B A O’ O Circle Circle ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Congruent Arcs 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 C A Congruent Arcs O’ O Y B D Circle Circle ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Concentric Circles Concentric Circles are two circles in the same plane with the same center but different radii A O Concentric Circles X ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Theorems ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Congruent Radii In the same or congruent circles all radii are congruent If C  O, r, s and t are radii, then r = s = t s r O C t Congruent Radii ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Central Angles
L25_Circles Congruent Arcs 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, then the intercepted arcs AB  XY C B A O Y X Congruent Central Angles = Congruent Arcs ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Congruent Central Angles 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  XOY C B A O Y X Congruent Arcs = Congruent Central Angles ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Arcs & Central Angles
L25_Circles Congruent Arcs & Central Angles In the same or in congruent circles, two arcs are congruent if and only if their central angles are congruent The arcs AB  XY, if and only if their central angles ACB  XOY C B A O Y X Congruent Arcs = Congruent Central Angles ERHS Math Geometry Mr. Chin-Sung Lin

Arc Addition Postulate
L25_Circles Arc Addition Postulate 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

L25_Circles Congruent Chords In the same or in congruent circles, if two central angles are congruent, then the chords are congruent If central angles ACB  XOY, then the chords AB  XY C B A O Y X Congruent Central Angles = Congruent Chords ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Congruent Central Angles 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  XOY C B A O Y X Congruent Chords = Congruent Central Angles ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Chords & Central Angles
L25_Circles Congruent Chords & Central Angles 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 their central angles ACB  XOY C B A O Y X Congruent Chords = Congruent Central Angles ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Congruent Chords 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 C B A O Y X Congruent Arcs = Congruent Chords ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Chords = Congruent Arcs
L25_Circles Congruent Arcs In the same or in congruent circles, if two chords are congruent, then their arcs are congruent If the chords AB  XY, then their arcs AB  XY C B A O Y X Congruent Chords = Congruent Arcs ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Arcs & Chords
L25_Circles Congruent Arcs & Chords 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 C B A O Y X Congruent Arcs = Congruent Chords ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Semicircles Postulate
L25_Circles Congruent Semicircles Postulate The diameter of a circle divides the circle into two congruent arcs (semicircles) If AB is a diameter of circle C, then APB  AQB P Diameter B A C Q ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Exercise 1 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

L25_Circles Exercise 2 Circle C has central angle ACB = 60o, DCE = 60o, and BCD = 170o, what’s the measure of the arc AD and BE? A D C E B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Exercise 3 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

L25_Circles Exercise 4 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 X C A Y 5 B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Exercise 5 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? C B A ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Exercise 6 If two concentric circles have radii 10 and 6 respectively, what’s the total area of the blue regions? C 10 6 ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents
Theorem of Chords ERHS Math Geometry Mr. Chin-Sung Lin

Chord Bisecting Theorem
L26_Chords Inscribed Angles and Tangents Chord Bisecting Theorem If a diameter is perpendicular to a chord, then it bisects the chord and its major and minor arcs Given: Diameter CD  AB Prove: 1) CD bisects AB 2) CD bisects AB and ACB O A B C D M Circle ERHS Math Geometry Mr. Chin-Sung Lin

Chord Bisecting Theorem
L26_Chords Inscribed Angles and Tangents Chord Bisecting Theorem If a diameter is perpendicular to a chord, then it bisects the chord and its major and minor arcs Given: Diameter CD  AB Prove: 1) CD bisects AB 2) CD bisects AB and ACB O A B C D M Circle 1 2 ERHS Math Geometry Mr. Chin-Sung Lin

Secants, Tangents, and Inscribed Angles
L25_Circles Secants, Tangents, and Inscribed Angles ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Secant A secant is a segment or line which passes through a circle, intersecting at two points A B Secant D C ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Tangent 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 Tangent C A ERHS Math Geometry Mr. Chin-Sung Lin

Degrees/Radians of a Circle
L25_Circles Degrees/Radians of a Circle There are 360 degrees in a circle or 2 radians in a circle Thus 2 radians equals 360 degrees 360o or 2 C A ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Inscribed Angle An inscribed angle is an angle that has its vertex and its sides contained in the chords of the circle (e.g., ADB) D A Inscribed Angle C B ERHS Math Geometry Mr. Chin-Sung Lin

L25_Circles Inscribed Polygon An inscribed polygon is a polygon whose vertices are on the circle Z W Inscribed Polygon C X Y ERHS Math Geometry Mr. Chin-Sung Lin

Circumscribed Polygon
L25_Circles Circumscribed Polygon Circumscribed polygon is a polygon whose sides are tangent to a circle Z W Circumscribed Polygon C X Y ERHS Math Geometry Mr. Chin-Sung Lin

Theorems of Inscribed Angles
L26_Chords Inscribed Angles and Tangents Theorems of Inscribed Angles ERHS Math Geometry Mr. Chin-Sung Lin

Inscribed Angle Theorem
L26_Chords Inscribed Angles and Tangents Inscribed Angle Theorem The measure of an inscribed angle is equal to one-half the measure of its intercepted arc Given: Inscribed angle ACB Prove: mACB = (1/2) m AB C Circle O A B ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Inscribed Angle Theorem The measure of an inscribed angle is equal to one-half the measure of its intercepted arc Given: Inscribed angle ACB Prove: mACB = (1/2) m AB Proof: (Case 1) Inscribed angles where one chord is a diameter C 2 O 1 3 A B ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Inscribed Angle Theorem The measure of an inscribed angle is equal to one-half the measure of its intercepted arc Given: Inscribed angle ACB Prove: mACB = (1/2) m AB Proof: (Case 2) Inscribed angles with the center of the circle in their interior C Circle 3 4 O 1 2 A B ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Inscribed Angle Theorem The measure of an inscribed angle is equal to one-half the measure of its intercepted arc Given: Inscribed angle ACB Prove: mACB = (1/2) m AB Proof: (Case 3) Inscribed angles with the center of the circle in their exterior Circle C 5 O 4 D 1 2 6 3 A B ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Inscribed Angle Theorem Given: Inscribed angle ACB Prove: mACB = (1/2) m AB m1 = m3 - m2 m4 = m6 - m5 m3 = 2 m6 m2 = 2 m5 m3 - m2 = 2 (m6 - m5) = 2 m4 m1 = 2 m4 m4 = (1/2) m1 mACB = (1/2) m AB D O A B C 1 2 3 4 5 6 Circle ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Inscribed Angle Theorem
L26_Chords Inscribed Angles and Tangents Congruent Inscribed Angle Theorem In the same or in congruent circles, if two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent Given: Inscribed angle ACB and ADB Prove: ACB  ADB C Circle D O A B ERHS Math Geometry Mr. Chin-Sung Lin

Right Inscribed Angle Theorem
L26_Chords Inscribed Angles and Tangents Right Inscribed Angle Theorem An angle inscribed in a semi-circle is a right angle Given: Inscribed angle ACB and AB is a diameter Prove: mACB = 90o C Circle A O B ERHS Math Geometry Mr. Chin-Sung Lin

Right Inscribed Angle Theorem
L26_Chords Inscribed Angles and Tangents Right Inscribed Angle Theorem An angle inscribed in a semi-circle is a right angle Given: Inscribed angle ACB and AB is a diameter Prove: mACB = 90o C Circle A O 180o B ERHS Math Geometry Mr. Chin-Sung Lin

Supplementary Inscribed Angle Theorem
L26_Chords Inscribed Angles and Tangents Supplementary Inscribed Angle Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary Given: ABCD is an inscribed quadrilateral of circle O Prove: mB + mD= 180 B Circle C O A D ERHS Math Geometry Mr. Chin-Sung Lin

Parallel Chords and Arcs Theorem
L26_Chords Inscribed Angles and Tangents Parallel Chords and Arcs Theorem In a circle, parallel chords intercept congruent arcs between them Given: AB || CD Prove: AC  BD Circle C D O A B ERHS Math Geometry Mr. Chin-Sung Lin

Exercises ERHS Math Geometry Mr. Chin-Sung Lin

Exercise C has an inscribed quadrilateral ABCD where A = 70o and B = 80o. What’s the measures of C and D? B 80o C O 70o A D ERHS Math Geometry Mr. Chin-Sung Lin

Exercise C has an inscribed quadrilateral ABCD where A = 70o and B = 80o. What’s the measures of C and D? B 80o C 110o O 70o A 100o D ERHS Math Geometry Mr. Chin-Sung Lin

Exercise C has an inscribed angle ADB = 30o, DB is the diameter. DEA =? E D 30o C B A Mr. Chin-Sung Lin

Exercise C has an inscribed angle ADB = 30o, DB is the diameter. DEA =? E 60o D 30o C 60o B A Mr. Chin-Sung Lin

Theorems of Tangents ERHS Math Geometry Mr. Chin-Sung Lin

Unique Tangent Postulate
L26_Chords Inscribed Angles and Tangents Unique Tangent Postulate At a given point on a circle, there is one and only one tangent to the circle Given P is on the circle O There is only one tangent AP to circle O Circle P O Tangent A ERHS Math Geometry Mr. Chin-Sung Lin

Perpendicular-Tangent Theorem
L26_Chords Inscribed Angles and Tangents Perpendicular-Tangent Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of contact Given: AB is a tangent to O P is the point of tangency Prove: AB  OP B Circle P O A ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Perpendicular-Tangent Theorem Given: AB is a tangent to O P is the point of tangency Prove: AB  OP (Indirect Proof) 1. Suppose OP is NOT perpendicular to AB 2. Draw a point D on AB, OD  AB 3. Draw point E on AB, PD = DE and E is on different side of D 4. ODP = ODE = 90° 5. OD = OD (Reflexive) E B Circle D P O A ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Perpendicular-Tangent Theorem Given: AB is a tangent to O P is the point of tangency Prove: AB  OP (Indirect Proof) 6. ODP  ODE (SAS) 7. OP = OE (CPCTC) 8. E is on O (by 7) 9. AB intersects the circle at two different points, so AB is not a tangent (contradicts to the given) 10. AB  OP (the opposite of the assumption is true) E B Circle D P O A ERHS Math Geometry Mr. Chin-Sung Lin

Converse of Perpendicular Tangent Theorem
L26_Chords Inscribed Angles and Tangents Converse of Perpendicular Tangent Theorem If a line is perpendicular to a radius at its outer endpoint, then it is a tangent to the circle Given: OP is a radius of O and AB  OP at P Prove: AB is a tangent to O B Circle P O A ERHS Math Geometry Mr. Chin-Sung Lin

Converse of Perpendicular Tangent Theorem
L26_Chords Inscribed Angles and Tangents Converse of Perpendicular Tangent Theorem Given: OP is a radius of O and AB  OP at P Prove: AB is a tangent to O 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) 5. AB is a tangent to O (Def. of circle) B D Circle P O A ERHS Math Geometry Mr. Chin-Sung Lin

L26_Chords Inscribed Angles and Tangents Perpendicular-Tangent Theorem 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

Common Tangents A common tangent is a line that is tangent to each of two circles A B B O O’ O O’ A Common Internal Tangent Common External Tangent ERHS Math Geometry Mr. Chin-Sung Lin

Common Tangents Two circles can have four, three, two, one, or no common tangents 4 3 2 1 ERHS Math Geometry Mr. Chin-Sung Lin

Circles Tangent Internally/Externally
L26_Chords Inscribed Angles and Tangents Circles Tangent Internally/Externally Two circles are said to be tangent to each other if they are tangent to the same line at the same point Tangent Externally Tangent Internally ERHS Math Geometry Mr. Chin-Sung Lin

Tangent Segments 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 Segments P Q ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Tangents Theorem
L26_Chords Inscribed Angles and Tangents Congruent Tangents Theorem If two tangents are drawn to a circle from the same external point, then these tangent segments are congruent Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AP  AQ P O A Q ERHS Math Geometry Mr. Chin-Sung Lin

Congruent Tangents Theorem
L26_Chords Inscribed Angles and Tangents Congruent Tangents Theorem If two tangents are drawn to a circle from the same external point, then these tangent segments are congruent Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AP  AQ (HL Postulate) P O A Q ERHS Math Geometry Mr. Chin-Sung Lin

Angles formed by Tangents Theorem
L26_Chords Inscribed Angles and Tangents Angles formed by Tangents Theorem 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 Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AO bisects PAQ P O A Q ERHS Math Geometry Mr. Chin-Sung Lin

Angles formed by Tangents Theorem
L26_Chords Inscribed Angles and Tangents Angles formed by Tangents Theorem 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. Given: AP and AQ are tangents to O, P and Q are points of tangency Prove: AO bisects POQ P O A Q ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 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

Exercise 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 (c) O’B = 9/2 B C O O’ A ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 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 D C 12 ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 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 4√ 5 8.9 B 8 A D C 12 ERHS Math Geometry Mr. Chin-Sung Lin

Exercise C has a tangent AB and a secant AE. If the diameter of the circle is 10 and AD = 8. AB = ? B A 10 8 D E C ERHS Math Geometry Mr. Chin-Sung Lin

Exercise C has a tangent AB and a secant AE. If the diameter of the circle is 10 and AD = 8. AB = ? B 5 A 8 5 D E C ERHS Math Geometry Mr. Chin-Sung Lin

Circumscribed Polygon &
L25_Circles Exercise Find the perimeter of the quadrilateral WXYZ Z D W 4 Circumscribed Polygon & Inscribed Circle A 8 C C 5 X B Y ERHS Math Geometry Mr. Chin-Sung Lin

Circumscribed Polygon &
L25_Circles Exercise Find the perimeter of the quadrilateral WXYZ Perimeter: 34 Z D W 4 Circumscribed Polygon & Inscribed Circle A 8 C C 5 X B Y ERHS Math Geometry Mr. Chin-Sung Lin

Angle Measurement Theorems
L27_Angle Measurement Theorems Angle Measurement Theorems ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Angle Measurement Theorems 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

An Angle Formed by A Tangent and A Chord
L27_Angle Measurement Theorems An Angle Formed by A Tangent and A Chord ERHS Math Geometry Mr. Chin-Sung Lin

Tangent-Chord Angle Theorem
L27_Angle Measurement Theorems Tangent-Chord Angle Theorem 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 Prove: 1) mABC = (1/2) m AB 2) mABD = (1/2) m AEB E A O C B D ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Chord Angle Theorem Given: CD is a tangent to O, B is the point of tangency, and AB is a chord Prove: 1) mABC = (1/2) m AB 2) mABD = (1/2) m AEB E A 1 O 2 C B D ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Chord Angle Theorem 1. Draw OA and OB, form 1 and 2 2. OB  CD 3. mABC + m2 = 90 4. OA = OB 5. m1 = m2 6. m1 + m2 + mAOB = 180 7. 2m2 + mAOB = 180 8. m2 + (1/2) mAOB = 90 9. m2 + (1/2) mAOB = mABC + m2 10. (1/2) mAOB = mABC 11. mABC = (1/2) m AB mABC = (1/2) (360 - m AB) 13. mABD = (1/2) m AEB E A 1 O 2 C B D ERHS Math Geometry Mr. Chin-Sung Lin

Tangent-Chord Angle Example
L27_Angle Measurement Theorems Tangent-Chord Angle Example 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 ? O A B C D E 70o 80o ERHS Math Geometry Mr. Chin-Sung Lin

Tangent-Chord Angle Example
L27_Angle Measurement Theorems Tangent-Chord Angle Example 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 ? O A B C D E 70o 80o 140o 160o 60o ERHS Math Geometry Mr. Chin-Sung Lin

Angles Formed by Two Tangents, Two Secants and A Secant A Tangent
L27_Angle Measurement Theorems Angles Formed by Two Tangents, Two Secants and A Secant A Tangent ERHS Math Geometry Mr. Chin-Sung Lin

Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems
L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems 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 O B A C D O B A C E O B A C D ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) O B A C D O B A C E E O B A C D E ERHS Math Geometry Mr. Chin-Sung Lin

Tangent-Tangent Angle Theorem (1)
L27_Angle Measurement Theorems Tangent-Tangent Angle Theorem (1) Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) 1. Draw BC, form 1 and 2 2. AB = AC 3. m1 = m2 = (1/2) m BC 4. mA + m1 + m2 = 180 5. mA + m BC = 180 6. m BC + m BEC = 360 7. (1/2) m BC + (1/2) m BEC = 180 8. mA + m BC = (1/2) m BC + (1/2) m BEC 9. mA = (-1/2) m BC + (1/2) m BEC 10. mA = (1/2) (m BEC - m BC ) O B A C E 1 2 ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent Angle Theorem (2) Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) 1. Draw OB and OC, form 1 and 2 2. OB  AB, OC  AC 3. mA + m1 + m2 + mBOC = 360 4. mA mBOC = 360 5. mA + mBOC = 180 6. mA + m BC = 180 7. m BC + m BEC = 360 8. (1/2) m BC + (1/2) m BEC = 180 9. mA + m BC = (1/2) m BC + (1/2) m BEC 10. mA = (-1/2) m BC + (1/2) m BEC 11. mA = (1/2) (m BEC - m BC ) B 1 O A 2 E C ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent Angle Theorem (3) Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC) 1. Draw BC, form 2 2. Extend AC, form 1 3. m2 = (1/2) m BC 4. m1 = (1/2) m BEC 5. mA = m1 - m2 6. mA = (1/2) m BEC - (1/2) m BC 7. mA = (1/2) (m BEC - m BC ) O B A C E 1 2 D ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC) O B A C D O B A C E E O B A C D E ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Secant Angle Theorem (1)
L27_Angle Measurement Theorems Secant-Secant Angle Theorem (1) D Given: O with secants AB and AC Prove: mA = (1/2) (m DE - m BC) 1. Draw DC 2. m2 = mA + m1 3. mA = m2 - m1 4. m2 = (1/2) m DE, m1 = (1/2) m BC 5. mA = (1/2) (m DE - m BC) B 1 O A 2 C E ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Secant Angle Theorem (2-1)
L27_Angle Measurement Theorems Secant-Secant Angle Theorem (2-1) Given: O with secants AB and AC Prove: mA = (1/2) (m DE - m BC) 1. Draw OB, OC, OD and OE 2. OB = OC = OD = OE 3. m3 = m4, m7 = m8 4. m5 = m3, m9 = m7 5. m3 = mBOA + mBAO 6. m7 = mCOA + mCAO 7. m5 + m9 = m m7 = (m3 + m7) A C E 1 2 B O D 3 4 5 7 8 9 ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Secant Angle Theorem (2-2)
L27_Angle Measurement Theorems Secant-Secant Angle Theorem (2-2) Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC) 8. m5 + m9 = (mBOA + mBAO + mCOA + mCAO) = (mBOC + mA) 9. m5 + m9 + 2(mBOC + mA) = 360 10. m5 + m9 + mBOC + mDOE = 360 11. mBOC - mDOE + 2mA = 0 12. 2 mA = mDOE - mBOC 13. mA = (1/2) (mDOE- mBOC) 14. mA = (1/2) (m DE- m BC) A C E 1 2 B O D 3 4 5 7 8 9 ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems Given: O with a secant AD and a tangent AC Prove: mA = (1/2) (m DEC - m BC) O B A C D O B A C E E O B A C D E ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Tangent Angle Theorem
L27_Angle Measurement Theorems Secant-Tangent Angle Theorem Given: O with a secant AD and a tangent AC Prove: mA = (1/2) (m DEC - m BC) 1. Draw BC 2. m2 = mA + m1 3. mA = m2 - m1 4. m2 = (1/2) m DEC, m1 = (1/2) m BC 5. mA = (1/2) (m DEC - m BC) D B 2 O A 1 E C ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems 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 O B A C D O B A C E O B A C D ERHS Math Geometry Mr. Chin-Sung Lin

Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 1
L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 1 If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG D B 40o F O A 50o C E G ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 1 If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG D B 40o F 65o O 130o A 50o 65o C E 25o G ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 2 If O with a tangent AB, secants AD, GB and GD, calculate m BD, and m BF B D H O 20o F A 40o C E 30o G ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 2 If O with a tangent AB, secants AD, GB and GD, calculate m BD, and m BF B 100o D 60o H O 20o F A 40o C E 30o G ERHS Math Geometry Mr. Chin-Sung Lin

An Angle Formed by Two Chords
L27_Angle Measurement Theorems An Angle Formed by Two Chords ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Angle Theorem
L27_Angle Measurement Theorems Chord-Chord Angle Theorem 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: mAMC = mBMD = (1/2) (m AC + m BD) O A B C D M ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Angle Theorem
L27_Angle Measurement Theorems Chord-Chord Angle Theorem Given: O with chords AB and CD Prove: mAMC = mBMD = (1/2) (m AC + m BD) 1. Draw BC 2. mAMC = mBMD 3. mAMC = m1 + m2 4. m1 = (1/2) m AC 5. m2 = (1/2) m BD 6. mAMC = (1/2) m AC + (1/2) m BD 7. mAMC = mBMD = (1/2) (m AC + m BD) O A B C D M 1 2 ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Angle Example
L27_Angle Measurement Theorems Chord-Chord Angle Example If O with chords AB, CD, AC and BD, calculate m AC and m AD A D O 70o M 60o C B 90o ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Angle Example
L27_Angle Measurement Theorems Chord-Chord Angle Example If O with chords AB, CD, AC and BD, calculate m AC and m AD 130o A D O 80o 70o M 60o C B 90o ERHS Math Geometry Mr. Chin-Sung Lin

L27_Angle Measurement Theorems Angle Measurement Theorems 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

L27_Angle Measurement Theorems
Exercise 1 O has a tangent ED and two parallel chords CD and AB. If the inscribed angle DAB = 20o, Find CDE. O B A D 20o E C ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 1 O has a tangent ED and two parallel chords CD and AB. If the inscribed angle DAB = 20o, Find CDE. O B A D 20o E C 100o 50o 40o 40o ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 2 C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF = ? mA = ? B 120o A D C E F ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 2 C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF = ? mA = ? B 120o 30o 60o 30o A 60o 60o D 60o C E 60o F ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 3 O has two secants CA and CB. If AE = ED and mEAB = 65, find ECB = ? E A C 65o D O B ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 3 O has two secants CA and CB. If AE = ED and mEAB = 65, find ECB = ? E A C 65o 65o D O 25o 25o B ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 4 ABCDE is a regular pentagon inscribed in O and BG is a tangent. Find ABG and AFE. C B D O F G A E ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 4 ABCDE is a regular pentagon inscribed in O and BG is a tangent. Find ABG and AFE. C B D O 36o 72o F 72o G 108o 72o A E ERHS Math Geometry Mr. Chin-Sung Lin

Segment Measurement Theorems
L28_Segment Measurement Theorems Segment Measurement Theorems ERHS Math Geometry Mr. Chin-Sung Lin

Segment Measurement Theorems
L28_Segment Measurement Theorems Segment Measurement Theorems Measure segments formed by Two chords A secant and a tangent Two secants ERHS Math Geometry Mr. Chin-Sung Lin

Segments Formed by Two Chords
L28_Segment Measurement Theorems Segments Formed by Two Chords ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Segment Theorem
L28_Segment Measurement Theorems Chord-Chord Segment Theorem 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 Given: AB and CD are chords of O, two chords intersect at E Prove: AE · BE = CE · DE O A B C D E ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Chord-Chord Segment Theorem 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 Given: AB and CD are chords of O, two chords intersect at E Prove: AE · BE = CE · DE O A B C D E 1 2 3 4 ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Chord-Chord Segment Theorem Given: AB and CD are chords of O, two chords intersect at E Prove: AE · BE = CE · DE 1. Connect BC and AD 2. m1 = m2 (Congruent inscribed angles) 3. m3 = m4 (Congruent inscribed angles) 4. CBE ~ ADE (AA similarity) 5. AE/CE = DE/BE (Corresponding sides proportional) 6. AE · BE = CE · DE (Cross product) O A B C D E 1 2 3 4 ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Segment Example
L28_Segment Measurement Theorems Chord-Chord Segment Example If O with chords AB and CD, CD = 10, CM = 6, and AM = 8, calculate AB = ? O A B C D M 6 8 10 ERHS Math Geometry Mr. Chin-Sung Lin

Chord-Chord Segment Example
L28_Segment Measurement Theorems Chord-Chord Segment Example 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) BM = 24 / 8 = 3 AB = = 11 O A B C D M 6 8 10 3 4 ERHS Math Geometry Mr. Chin-Sung Lin

Segments Formed by A Secant and A Tangent
L28_Segment Measurement Theorems Segments Formed by A Secant and A Tangent ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Tangent Segment Theorem
L28_Segment Measurement Theorems Secant-Tangent Segment Theorem 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 is a tangent of O, Prove: AD · AB = AC2 A C B O D ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Secant-Tangent Segment Theorem 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 is a tangent of O, Prove: AD · AB = AC2 A C B O D 2 1 ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Secant-Tangent Segment Theorem Given: A is an external point to O, AD is a secant and AC is a tangent of O, Prove: AD · AB = AC2 1. Connect BC and CD 2. m1 = (1/2) m BC (Tangent-chord angles theorem) 3. m2 = (1/2) m BC (Inscribed angles theorem) 4. m1 = m2 (Substitution property) 5. mA = mA (Reflexive property) 6. CBA ~ DCA (AA similarity) 7. AB/AC = AC/AD (Corresponding sides proportional) 8. AD · AB = AC2 (Cross product) A C B O D 2 1 ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Tangent Segment Example
L28_Segment Measurement Theorems Secant-Tangent Segment Example If O with tangent AC and secant AD, OD = 5 and AB = 6, calculate AC = ? O B 5 D 6 A C ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Tangent Segment Example
L28_Segment Measurement Theorems Secant-Tangent Segment Example 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 O B 5 5 D 6 A 4 √6 C ERHS Math Geometry Mr. Chin-Sung Lin

Segments Formed by Two Secants
L28_Segment Measurement Theorems Segments Formed by Two Secants ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Secant Segment Theorem
L28_Segment Measurement Theorems Secant-Secant Segment Theorem 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 Given: A is an external point to O, AD and AE are secants to O Prove: AD · AB = AE · AC A C E B O D ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Secant-Secant Segment Theorem 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 Given: A is an external point to O, AD and AE are secants to O Prove: AD · AB = AE · AC A C E B O D 1 2 ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Secant-Secant Segment Theorem Given: A is an external point to O, AD and AE are secants to O Prove: AD · AB = AE · AC 1. Connect BE and CD 2. m1 = (1/2) m BC (Inscribed angles theorem) 3. m2 = (1/2) m BC (Inscribed angles theorem) 4. m1 = m2 (Substitution property) 5. mA = mA (Reflexive property) 6. EBA ~ DCA (AA similarity) 7. AD/AE = AC/AB (Corresponding sides proportional) 8. AD · AB = AE · AC (Cross product) A C E B O D 1 2 ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Secant Segment Example
L28_Segment Measurement Theorems Secant-Secant Segment Example If O with secants AC and AE, OC = DE = x, AD = 10 and AB = 8, calculate BC = ? O B x C 8 A 10 x D E ERHS Math Geometry Mr. Chin-Sung Lin

Secant-Secant Segment Example
L28_Segment Measurement Theorems Secant-Secant Segment Example 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 = x 3x = 18 X = 6 BC = 12 O B C 8 A 12 10 6 D E ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems
Exercise 1 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 A O 4 E B ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 1 O has a tangent AF and two secants AC and AB. If AD = 3, CD = 9, and AE = 4, find AF = ? BE = ? AF2 = AD · AC = 3 · (3 + 9) AF2 = 36 AF = 6 AF2 = AB · AE 36 = 4 (BE + 4) BE = 5 F C 6 9 3 D A O 4 5 E B ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 2 C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ? A 6 3 C M 3 E D P 2 N F ERHS Math Geometry Mr. Chin-Sung Lin

Exercise 2 C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ? AP · PF = DP · PE 6 · 2 = DP · 3 DP = 4 AC = 5 CN = ( )1/2 A 5 6 3 C M 5 3 E D P 4 2 N F ERHS Math Geometry Mr. Chin-Sung Lin

Circles in a Coordinate Plane
L26_Chords Inscribed Angles and Tangents Circles in a Coordinate Plane ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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) x O x x D (0, –5) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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? x2 + y2 = 52 or x2 + y2 = 25 y B (0, 5) P (x, y) y 5 y C (-5, 0) A (5, 0) x O x x D (0, –5) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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) x O x x D (0, –r) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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? x2 + y2 = r2 y B (0, r) P (x, y) y r y C (-r, 0) A (r, 0) x O x x D (0, –r) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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 5 |y -4| C (-3, 4) A (7, 4) (2, 4) x y = 4 |x – 2| D (2, –1) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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)2 + (y – 4)2 = 52 or (x – 2)2 + (y – 4)2 = 25 x = 2 B (2, 9) P (x, y) y 5 |y -4| C (-3, 4) A (7, 4) (2, 4) x y = 4 |x – 2| D (2, –1) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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 r |y -k| 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

L28_Segment Measurement Theorems Circles in a Coordinate Plane 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)2 + (y – k)2 = r2 x = h B (h, k+r) P (x, y) y r |y -k| 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

Equation of a Circle Center-radius equation of a circle with radius r and center (h, k) is (x – h)2 + (y – k)2 = r2 P (x, y) r (h, k) ERHS Math Geometry Mr. Chin-Sung Lin

Center of a Circle 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

Center of a Circle 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 C (h, k) = ( , ) P (x1, y1) r x1 + x2 y1 + y2 C (h, k) r Q (x2, y2) ERHS Math Geometry Mr. Chin-Sung Lin

Center of a Circle Example
L28_Segment Measurement Theorems Center of a Circle Example 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

Center of a Circle Example
L28_Segment Measurement Theorems Center of a Circle Example 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) C (h, k) = ( , ) = (2, 3) P (5, 7) 5 + (-1) (-1) r C (h, k) r Q (-1, -1) ERHS Math Geometry Mr. Chin-Sung Lin

Radius of a Circle 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

Radius of a Circle 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 r = ½ PQ = ½ √ (x2 – x1)2 + (y2 – y1)2 P (x1, y1) r C (h, k) r Q (x2, y2) ERHS Math Geometry Mr. Chin-Sung Lin

Radius of a Circle Example
L28_Segment Measurement Theorems Radius of a Circle Example 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

Radius of a Circle Example
L28_Segment Measurement Theorems Radius of a Circle Example 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? r = ½ PQ = ½ √ (-1 – 5)2 + (-1 – 7)2 = ½ (10) = 5 P (5, 7) r C (h, k) r Q (-1, -1) ERHS Math Geometry Mr. Chin-Sung Lin

Circles in a Coordinate Plane Exercise
L28_Segment Measurement Theorems Circles in a Coordinate Plane Exercise Write an equation of a circle with center at (3, -2) and radius of length 7 What are the coordinates of the endpoints of the horizontal diameter? ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane Exercise Write an equation of a circle with center at (3, -2) and radius of length 7 (x–3)2 + (y+2)2 = 49 What are the coordinates of the endpoints of the horizontal diameter? (10, -2), (-4, -2) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane Exercise A circle C has a diameter PQ with end-points at P (-2, 9) and Q (4, 1) What is the center (C) of the circle? What is the radius (r) of the circle? What is the equation of the circle? What are the coordinates of the endpoints of the horizontal diameter? What are the coordinates of the endpoints of the vertical diameter? What are the coordinates of two other points on the circle? ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane Exercise A circle C has a diameter PQ with end-points at P (-2, 9) and Q (4, 1) What is the center (C) of the circle? (1, 5) What is the radius (r) of the circle? 5 What is the equation of the circle? (x–1)2 + (y–5)2 = 25 What are the coordinates of the endpoints of the horizontal diameter? (-4, 5), (6, 5) What are the coordinates of the endpoints of the vertical diameter? (1, 10), (1, 0) What are the coordinates of two other points on the circle? (4, 9), (-2, 1) ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane Exercise Based on the diagram, write an equation of the circle Find the area of the circle ERHS Math Geometry Mr. Chin-Sung Lin

L28_Segment Measurement Theorems Circles in a Coordinate Plane Exercise Based on the diagram, write an equation of the circle Find the area of the circle (a) (x+4)2 + (y+4)2 = 25 (b) 25π ERHS Math Geometry Mr. Chin-Sung Lin

L6_Coordinate Geometry & Proofs
Q & A ERHS Math Geometry Mr. Chin-Sung Lin

L6_Coordinate Geometry & Proofs
The End ERHS Math Geometry Mr. Chin-Sung Lin

Eleanor Roosevelt High School Chin-Sung Lin

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