Eleanor Roosevelt High School Chin-Sung Lin. Mr. Chin-Sung Lin ERHS Math Geometry.
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Transcript of Eleanor Roosevelt High School Chin-Sung Lin. Mr. Chin-Sung Lin ERHS Math Geometry.
Geometry of The Circle
Eleanor Roosevelt High School Chin-Sung Lin
Geometry Chapter 13
Arcs, Angles, and Chords
Mr. Chin-Sung LinERHS Math Geometry
CircleA 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
CCircle
Mr. Chin-Sung LinERHS Math Geometry
CenterIt is the center of the circle and the distance from this
point to any other point on the circumference is the same
CCircle
Mr. Chin-Sung LinERHS Math Geometry
Center
RadiusA radius is the line segment connecting (sometimes
referred to as the “distance between”) the center and the circle itself
CCircle
Mr. Chin-Sung LinERHS Math Geometry
Center
rA
Radius
CircumferenceA 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 (2r)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
Center
rA
Radius
Circumference
ChordA chord is a line segment with endpoints on the
circle
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Chord
DiameterA diameter of a circle is a chord that has the center of
the circle as one of its points
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B ADiameter
ArcAn arc is a part of the circumference of a circle
(e.g., arc AB)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Arc
Central AngleA central angle is an angle in a circle with vertex at
the center of the circle
(e.g., ACB)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Arc
Major ArcGiven two points on a circle, the major arc is the
longest arc linking them
(e.g., arc ADB, mACB > 180)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Major Arc D
Minor ArcGiven two points on a circle, the minor arc is the
shortest arc linking them
(e.g., arc AB, mACB < 180)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Minor Arc
SemicircleHalf a circle. If the endpoints of an arc are the endpoints
of a diameter, then the arc is a semicircle
(e.g., arc ADB, mACB = 180)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B A
SemicircleD
Adjacent ArcsAdjacent arcs are non-overlapping arcs with the same
radius and center, sharing a common endpoint
(e.g., arc AB and AD)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Adjacent ArcsD
Intercepted ArcIntercepted Arc is the part of a circle that lies between
two lines that intersect it
(e.g., arc AB and XY)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Intercepted Arcs
X
Y
Arc LengthAn arc length is the distance along the curved line
making up the arc
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A
Arc Length
Degree Measure of an ArcThe degree measure of an arc is equal to the measure
of the central angle that intercepts the arc
(e.g., m AB = mACB)
CCircle
Mr. Chin-Sung LinERHS Math Geometry
B
A Measure of Central Angle
= Measure of Intercepted
Arc
Measure of a Minor ArcThe measure of minor arc is the degree measure of
central angle of the intercepted arc
(e.g., m AB = mACB)
Mr. Chin-Sung LinERHS Math Geometry
CCircle
B
ADegree Measure of a Minor Arc
Measure of a Major ArcThe measure of major arc is 360 minus the degree
measure of the minor arc
(e.g., m ADB = 360 – mACB)
Mr. Chin-Sung LinERHS Math Geometry
C Circle
B
ADegree Measure of a Major Arc
D
Congruent CirclesCongruent circles are circles that have congruent radii
(e.g., O ≅ O’)
Mr. Chin-Sung LinERHS Math Geometry
O
Circle
A
Congruent Circles
O’
Circle
B
Congruent ArcsCongruent arcs are arcs that have the same degree
measure and are in the same circle or in congruent circles (e.g., AB ≅ CD ≅ XY)
Mr. Chin-Sung LinERHS Math Geometry
O
Circle
ACongruen
t ArcsO’
Circle
X
BY
C
D
Concentric CirclesConcentric Circles are two circles in the same plane
with the same center but different radii
Mr. Chin-Sung LinERHS Math Geometry
OConcentric Circles
A
X
Theorems
Mr. Chin-Sung LinERHS Math Geometry
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
Mr. Chin-Sung LinERHS Math Geometry
Cr O
s
Congruent Radii
t
Congruent ArcsIn 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
Mr. Chin-Sung LinERHS Math Geometry
CB
A
OY
X
Congruent Central Angles = Congruent Arcs
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
Mr. Chin-Sung LinERHS Math Geometry
CB
A
OY
X
Congruent Arcs = Congruent Central Angles
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
Mr. Chin-Sung LinERHS Math Geometry
CB
A
OY
X
Congruent Arcs = Congruent Central Angles
Arc Addition PostulateIf AB and BC are two adjacent arcs of the same circle ,
then AB + BC = ABC and mAB + mBC = mABC
Mr. Chin-Sung LinERHS Math Geometry
OCircle
A
B
C
Congruent ChordsIn 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
Mr. Chin-Sung LinERHS Math Geometry
CB
A
OY
X
Congruent Central Angles = Congruent Chords
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
Mr. Chin-Sung LinERHS Math Geometry
Congruent Chords = Congruent Central Angles
CB
A
OY
X
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
Mr. Chin-Sung LinERHS Math Geometry
Congruent Chords = Congruent Central Angles
CB
A
OY
X
Congruent ChordsIn 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
Mr. Chin-Sung LinERHS Math Geometry
CB
A
OY
X
Congruent Arcs= Congruent Chords
Congruent ArcsIn 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
Mr. Chin-Sung LinERHS Math Geometry
Congruent Chords = Congruent Arcs
CB
A
OY
X
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
Mr. Chin-Sung LinERHS Math Geometry
CB
A
OY
X
Congruent Arcs= Congruent Chords
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
Mr. Chin-Sung LinERHS Math Geometry
C
Diameter AB
Q
P
Exercise 1 Circle C has central angle ACB = 60o, what’s the measure of
the arc ADB?
CB
AD
Mr. Chin-Sung LinERHS Math Geometry
Exercise 2 Circle C has central angle ACB = 60o, DCE = 60o, and BCD =
170o, what’s the measure of the arc AD and BE?
CB
AD
E
Mr. Chin-Sung LinERHS Math Geometry
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?
CB
AD
E
F
Mr. Chin-Sung LinERHS Math Geometry
Exercise 4The 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
5
C A
B
Y
X
Mr. Chin-Sung LinERHS Math Geometry
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?
CB
A
Mr. Chin-Sung LinERHS Math Geometry
Exercise 6If two concentric circles have radii 10 and 6 respectively, what’s
the total area of the blue regions?
C10
6
Mr. Chin-Sung LinERHS Math Geometry
Theorem of Chords
Mr. Chin-Sung LinERHS Math Geometry
Chord Bisecting TheoremIf a diameter is perpendicular to a chord, then it bisects
the chord and its major and minor arcs
Given: Diameter CD ABProve:
1) CD bisects AB2) CD bisects AB and ACB
Circle
O
A B
C
D
M
Mr. Chin-Sung LinERHS Math Geometry
Chord Bisecting TheoremIf a diameter is perpendicular to a chord, then it bisects
the chord and its major and minor arcs
Given: Diameter CD ABProve:
1) CD bisects AB2) CD bisects AB and ACB
Circle
O
A B
C
D
M
Mr. Chin-Sung LinERHS Math Geometry
1 2
Secants, Tangents, and Inscribed Angles
Mr. Chin-Sung LinERHS Math Geometry
SecantA secant is a segment or line which passes through a
circle, intersecting at two points
C
Secant
AB
D
Mr. Chin-Sung LinERHS Math Geometry
TangentA tangent is a line in the plane of a circle that intersects the
circle in exactly one point (called the point of tangency)
C Tangent
Point of TangentB
A
D
Mr. Chin-Sung LinERHS Math Geometry
Degrees/Radians of a CircleThere are 360 degrees in a circle or 2 radians in a circle
Thus 2 radians equals 360 degrees
360o or 2C A
Mr. Chin-Sung LinERHS Math Geometry
Inscribed AngleAn inscribed angle is an angle that has its vertex and its sides
contained in the chords of the circle
(e.g., ADB)
CB
AD
Mr. Chin-Sung LinERHS Math Geometry
Inscribed Angle
Inscribed PolygonAn inscribed polygon is a polygon whose vertices are on the
circle
Inscribed Polygon
CX
WZ
Y
Mr. Chin-Sung LinERHS Math Geometry
Circumscribed PolygonCircumscribed polygon is a polygon whose sides are tangent
to a circle
Circumscribed Polygon
X
WZ
C
Y
Mr. Chin-Sung LinERHS Math Geometry
Theorems of Inscribed Angles
Mr. Chin-Sung LinERHS Math Geometry
Inscribed Angle TheoremThe measure of an inscribed angle is equal to one-half
the measure of its intercepted arc
Given: Inscribed angle ACBProve: mACB = (1/2) m AB
Circle
O
A B
C
Mr. Chin-Sung LinERHS Math Geometry
Inscribed Angle TheoremThe measure of an inscribed angle is equal to one-half
the measure of its intercepted arc
Given: Inscribed angle ACBProve: mACB = (1/2) m AB
Proof: (Case 1)Inscribed angles where one chord is a diameter
1
O
A B
C
2
3
Mr. Chin-Sung LinERHS Math Geometry
Inscribed Angle TheoremThe measure of an inscribed angle is equal to one-half
the measure of its intercepted arc
Given: Inscribed angle ACBProve: mACB = (1/2) m AB
Proof: (Case 2)Inscribed angles with the center of the circle in their interior
Circle
O
A B
C
1 2
3 4
Mr. Chin-Sung LinERHS Math Geometry
Inscribed Angle TheoremThe measure of an inscribed angle is equal to one-half
the measure of its intercepted arc
Given: Inscribed angle ACBProve: mACB = (1/2) m AB
Proof: (Case 3)Inscribed angles with the center of the circle
in their exterior
Circle
O
A B
C
D 123
45
6
Mr. Chin-Sung LinERHS Math Geometry
Inscribed Angle TheoremGiven: Inscribed angle ACBProve: mACB = (1/2) m AB
m1 = m3 - m2m4 = m6 - m5m3 = 2 m6m2 = 2 m5 m3 - m2 = 2 (m6 - m5) = 2 m4m1 = 2 m4m4 = (1/2) m1 mACB = (1/2) m AB
Circle
D
O
A B
C
12
3
45
6
Mr. Chin-Sung LinERHS Math Geometry
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
Mr. Chin-Sung LinERHS Math Geometry
Circle
O
A B
C
D
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
Mr. Chin-Sung LinERHS Math Geometry
Circle
O
A B
C
D
Right Inscribed Angle TheoremAn angle inscribed in a semi-circle is a right angle
Given: Inscribed angle ACB and AB is a diameter
Prove: mACB = 90o
Circle
O
A
B
C
Mr. Chin-Sung LinERHS Math Geometry
Right Inscribed Angle TheoremAn angle inscribed in a semi-circle is a right angle
Given: Inscribed angle ACB and AB is a diameter
Prove: mACB = 90o
Circle
O
A
B
C
180o
Mr. Chin-Sung LinERHS Math Geometry
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: mB + mD= 180
Circle
O
A
B
C
D
Mr. Chin-Sung LinERHS Math Geometry
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: mB + mD= 180
Circle
O
A
B
C
D
Mr. Chin-Sung LinERHS Math Geometry
Parallel Chords and Arcs TheoremIn a circle, parallel chords intercept congruent arcs
between them
Given: AB || CDProve: AC BD
Circle
O
A B
C D
Mr. Chin-Sung LinERHS Math Geometry
Parallel Chords and Arcs TheoremIn a circle, parallel chords intercept congruent arcs
between them
Given: AB || CDProve: AC BD
Circle
O
A B
C D
Mr. Chin-Sung LinERHS Math Geometry
Exercises
Mr. Chin-Sung LinERHS Math Geometry
ExerciseC has an inscribed quadrilateral ABCD where A = 70o and B
= 80o. What’s the measures of C and D?
Mr. Chin-Sung LinERHS Math Geometry
70o
O
A
B
C
D
80o
ExerciseC has an inscribed quadrilateral ABCD where A = 70o and B
= 80o. What’s the measures of C and D?
Mr. Chin-Sung LinERHS Math Geometry
70o
O
A
B
C
D
80o
100o
110o
Exercise
Mr. Chin-Sung Lin
C has an inscribed angle ADB = 30o, DB is the diameter. DEA =?
CB
A
D
30o
E
Exercise
Mr. Chin-Sung Lin
C has an inscribed angle ADB = 30o, DB is the diameter. DEA =?
CB
A
D
30o
E
60o
60o
Theorems of Tangents
Mr. Chin-Sung LinERHS Math Geometry
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 OThere is only one tangent APto circle O
Circle
O
A
P
Tangent
Mr. Chin-Sung LinERHS Math Geometry
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 OP is the point of tangency
Prove: AB OP
Circle
O
A
P
B
Mr. Chin-Sung LinERHS Math Geometry
Perpendicular-Tangent TheoremGiven: AB is a tangent to O
P is the point of tangencyProve: 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)
Circle
O
A
P
BD
E
Mr. Chin-Sung LinERHS Math Geometry
Perpendicular-Tangent TheoremGiven: AB is a tangent to O
P is the point of tangencyProve: 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)
Circle
O
A
P
BD
E
Mr. Chin-Sung LinERHS Math Geometry
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 andAB OP at P
Prove: AB is a tangent to O
Circle
OA
P
Converse of Perpendicular Tangent Theorem
B
Mr. Chin-Sung LinERHS Math Geometry
Converse of Perpendicular Tangent Theorem
Given: OP is a radius of O andAB 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)
Circle
O
A
P
BD
Mr. Chin-Sung LinERHS Math Geometry
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
Circle
O
A
P
B
Mr. Chin-Sung LinERHS Math Geometry
Common Tangents
A common tangent is a line that is tangent to each of two circles
O
Mr. Chin-Sung LinERHS Math Geometry
O’
A
B
Common Internal Tangent
O O’
A B
Common External Tangent
Common Tangents
Two circles can have four, three, two, one, or no common tangents
Mr. Chin-Sung LinERHS Math Geometry
01234
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
Mr. Chin-Sung LinERHS Math Geometry
Tangent InternallyTangent Externally
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
O
P
R
TangentSegments
Mr. Chin-Sung LinERHS Math Geometry
Q
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
O
A
P
Q
Mr. Chin-Sung LinERHS Math Geometry
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)
O
A
P
Q
Mr. Chin-Sung LinERHS Math Geometry
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
O
A
P
Q
Mr. Chin-Sung LinERHS Math Geometry
Angles formed by Tangents TheoremIf 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
O
A
P
Q
Mr. Chin-Sung LinERHS Math Geometry
Exercises
Mr. Chin-Sung LinERHS Math Geometry
ExerciseCircles 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
O
Mr. Chin-Sung LinERHS Math Geometry
O’
A
B
C
ExerciseCircles 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
O
Mr. Chin-Sung LinERHS Math Geometry
O’
A
B
C
ExerciseC 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
C
B
A
12D
8
Mr. Chin-Sung LinERHS Math Geometry
ExerciseC 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
(a) 4√ 5
(b) 8.9C
B
A
12D
8
Mr. Chin-Sung LinERHS Math Geometry
ExerciseC has a tangent AB and a secant AE. If the diameter of the circle is
10 and AD = 8. AB = ?
C
B
A
8D10
E
Mr. Chin-Sung LinERHS Math Geometry
ExerciseC has a tangent AB and a secant AE. If the diameter of the circle is
10 and AD = 8. AB = ?
C
A
8D
5
E5
Mr. Chin-Sung LinERHS Math Geometry
B
ExerciseFind the perimeter of the quadrilateral WXYZ
8
Circumscribed Polygon &
Inscribed Circle
X
WZ
C
Y
Mr. Chin-Sung LinERHS Math Geometry
A
B
C
D
5
4
ExerciseFind the perimeter of the quadrilateral WXYZ
Perimeter: 34 8
Circumscribed Polygon &
Inscribed Circle
X
WZ
C
Y
Mr. Chin-Sung LinERHS Math Geometry
A
B
C
D
5
4
Angle Measurement Theorems
Mr. Chin-Sung LinERHS Math Geometry
Angle Measurement Theorems
Measure an angle formed by
A tangent and a chord
Two tangents
Two secants
A tangent and a secant
Two chords
Mr. Chin-Sung LinERHS Math Geometry
An Angle Formed by A Tangent and A Chord
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Chord Angle TheoremThe 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) mABC = (1/2) m AB2) mABD = (1/2) m AEB
O
A
BC D
E
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Chord Angle TheoremGiven: CD is a tangent to O, B is the point of tangency, and
AB is a chord
Prove:1) mABC = (1/2) m AB2) mABD = (1/2) m AEB 1 O
A
BC D
2
E
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Chord Angle Theorem
1. Draw OA and OB, form 1 and 22. OB CD3. mABC + m2 = 904. 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 + m210. (1/2) mAOB = mABC 11. mABC = (1/2) m AB12. 180 - mABC = (1/2) (360 - m AB)13. mABD = (1/2) m AEB
Mr. Chin-Sung LinERHS Math Geometry
1 O
A
BC D
2
E
Tangent-Chord Angle ExampleIf 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
BC D
E70o
80o
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Chord Angle ExampleIf 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
BC D
E70o
80o
70o 80o
140o160o
60o
Mr. Chin-Sung LinERHS Math Geometry
Angles Formed by Two Tangents, Two Secants
and A Secant A Tangent
Mr. Chin-Sung LinERHS Math Geometry
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
CE
OB
A
C
D
OB
A
C
D
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems
Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC)
E
OB
A
C
D
OB
A
C
D
O
B
A
CE
E
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent Angle Theorem (1)Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC)
1. Draw BC, form 1 and 2
2. AB = AC
3. m1 = m2 = (1/2) m BC
4. mA + m1 + m2 = 180
5. mA + m BC = 180
6. m BC + m BEC = 360
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 )
O
B
A
CE
1
2
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent Angle Theorem (2)Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC)
1. Draw OB and OC, form 1 and 22. OB AB, OC AC3. mA + m1 + m2 + mBOC = 3604. mA + 90 + 90 + mBOC = 3605. mA + mBOC = 180 6. mA + m BC = 1807. 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 )
O
B
A
CE
1
2
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent Angle Theorem (3)Given: O with tangents AB and AC Prove: mA = (1/2) (m BEC - m BC)
1. Draw BC, form 2
2. Extend AC, form 1
3. m2 = (1/2) m BC
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 )
O
B
A
CE1
2
Mr. Chin-Sung LinERHS Math Geometry
D
Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems
Given: O with secants AD and AE Prove: mA = (1/2) (m DE - m BC)
E
OB
A
C
D
OB
A
C
D
O
B
A
CE
E
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Angle Theorem (1)Given: O with secants AB and AC
Prove: mA = (1/2) (m DE - m BC)
1. Draw DC
2. m2 = mA + m1
3. mA = m2 - m1
4. m2 = (1/2) m DE, m1 = (1/2) m BC
5. mA = (1/2) (m DE - m BC)
A
CE
B
O
D
1
2
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Angle Theorem (2-1)Given: O with secants AB and AC Prove: mA = (1/2) (m DE - m BC)
1. Draw OB, OC, OD and OE
2. OB = OC = OD = OE
3. m3 = m4, m7 = m8
4. m5 = 180 - 2 m3, m9 = 180 - 2 m7
5. m3 = mBOA + mBAO
6. m7 = mCOA + mCAO
7. m5 + m9 = 180 - 2 m3 + 180 - 2 m7
= 360 - 2(m3 + m7)
A
CE
1
2
B
O
D
34
5
78
9
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Angle Theorem (2-2)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
10. m5 + m9 + mBOC + mDOE = 360
11. mBOC - mDOE + 2mA = 0
12. 2 mA = mDOE - mBOC
13. mA = (1/2) (mDOE- mBOC)
14. mA = (1/2) (m DE- m BC)
A
CE
1
2
B
O
D
34
5
78
9
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Theorems
Given: O with a secant AD and a tangent AC Prove: mA = (1/2) (m DEC - m BC)
E
OB
A
C
D
OB
A
C
D
O
B
A
CE
E
Mr. Chin-Sung LinERHS Math Geometry
Secant-Tangent Angle Theorem
Given: O with a secant AD and a tangent AC
Prove: mA = (1/2) (m DEC - m BC)
1. Draw BC
2. m2 = mA + m1
3. mA = m2 - m1
4. m2 = (1/2) m DEC, m1 = (1/2) m BC
5. mA = (1/2) (m DEC - m BC)
A
CE
1
B
O
D
2
Mr. Chin-Sung LinERHS Math Geometry
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
CE
OB
A
C
D
OB
A
C
D
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 1
If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG
O
B
A
C E
G
D40o
50o
F
Mr. Chin-Sung LinERHS Math Geometry
Tangent-Tangent, Secant-Secant, Secant-Tangent Angle Example 1
If O with tangents AD, AC, secants GB and GD, calculate m BC, and mG
O
B
A
C E
G
D40o
50o
F
65o
65o
130o
25o
Mr. Chin-Sung LinERHS Math Geometry
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
O
B
A
C E
H
G
D
F
30o
40o
20o
Mr. Chin-Sung LinERHS Math Geometry
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
O
B
A
C E
H
G
D
F
30o
40o
20o
100o
60o
Mr. Chin-Sung LinERHS Math Geometry
An Angle Formed by Two Chords
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Angle TheoremThe 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)
Mr. Chin-Sung LinERHS Math Geometry
O
A
BC
D
M
Chord-Chord Angle TheoremGiven: O with chords AB and CD Prove: mAMC = mBMD = (1/2) (m AC + m BD)
1. Draw BC
2. mAMC = mBMD 3. mAMC = m1 + m24. m1 = (1/2) m AC5. m2 = (1/2) m BD 6. mAMC = (1/2) m AC + (1/2) m BD 7. mAMC = mBMD = (1/2) (m AC + m BD)
O
A
BC
D
M12
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Angle ExampleIf O with chords AB, CD, AC and BD, calculate m AC and
m AD
O
A
BC
D
M70o
90o
60o
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Angle ExampleIf O with chords AB, CD, AC and BD, calculate m AC and
m AD
O
A
BC
D
M70o
90o
60o
80o
130o
Mr. Chin-Sung LinERHS Math Geometry
Angle Measurement Theorems
Measure an angle formed by
A tangent and a chord
Two tangents
Two secants
A tangent and a secant
Two chords
Mr. Chin-Sung LinERHS Math Geometry
Exercises
Mr. Chin-Sung LinERHS Math Geometry
Exercise 1 O has a tangent ED and two parallel chords CD and AB. If the
inscribed angle DAB = 20o, Find CDE.
OBA
D
20o
E
C
Mr. Chin-Sung LinERHS Math Geometry
Exercise 1 O has a tangent ED and two parallel chords CD and AB. If the
inscribed angle DAB = 20o, Find CDE.
OBA
D
20o
E
C
40o40o
100o
50o
Mr. Chin-Sung LinERHS Math Geometry
Exercise 2 C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF
= ? mA = ?
C
B
A
DE
F
120o
Mr. Chin-Sung LinERHS Math Geometry
Exercise 2 C has a tangent AB and a secant AE. If m BE = 120, m BD = ? m EF
= ? mA = ?
C
B
A
30o
DE
F
60o
120o
60o
60o
60o
60o
30o
Mr. Chin-Sung LinERHS Math Geometry
Exercise 3 O has two secants CA and CB. If AE = ED and mEAB = 65, find
ECB = ?
C
B
A
D
65o
O
E
Mr. Chin-Sung LinERHS Math Geometry
Exercise 3 O has two secants CA and CB. If AE = ED and mEAB = 65, find
ECB = ?
C
B
A
D
65o
O
E
25o 25o
65o
Mr. Chin-Sung LinERHS Math Geometry
Exercise 4 ABCDE is a regular pentagon inscribed in O and BG is a
tangent. Find ABG and AFE.C
B
A
D
O
E
FG
Mr. Chin-Sung LinERHS Math Geometry
Exercise 4 ABCDE is a regular pentagon inscribed in O and BG is a
tangent. Find ABG and AFE.C
B
A
D
O
E
FG
72o
36o
72o72o
108o
Mr. Chin-Sung LinERHS Math Geometry
Segment Measurement Theorems
Mr. Chin-Sung LinERHS Math Geometry
Segment Measurement Theorems
Measure segments formed by
Two chords
A secant and a tangent
Two secants
Mr. Chin-Sung LinERHS Math Geometry
Segments Formed by Two Chords
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Segment TheoremIf 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 · DEO
A
B
C
DE
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Segment TheoremIf 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 · DEO
A
B
C
DE
1
2 3
4
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Segment TheoremGiven: AB and CD are chords of O,
two chords intersect at E Prove: AE · BE = CE · DE1. Connect BC and AD
2. m1 = m2 (Congruent inscribed angles)
3. m3 = m4 (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
DE
1
2 3
4
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Segment ExampleIf O with chords AB and CD, CD = 10, CM = 6, and AM = 8,
calculate AB = ?
O
A
BC
D
M
6
8
10
Mr. Chin-Sung LinERHS Math Geometry
Chord-Chord Segment ExampleIf 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 = 3 + 8 = 11
O
A
BC
D
M
6
8
103
4
Mr. Chin-Sung LinERHS Math Geometry
Segments Formed by A Secant and A
Tangent
Mr. Chin-Sung LinERHS Math Geometry
Secant-Tangent Segment TheoremIf 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
Mr. Chin-Sung LinERHS Math Geometry
Secant-Tangent Segment TheoremIf 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
Mr. Chin-Sung LinERHS Math Geometry
Secant-Tangent Segment TheoremGiven: 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. m1 = (1/2) m BC (Tangent-chord angles theorem)
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)
A
C
B
O
D
2
1
Mr. Chin-Sung LinERHS Math Geometry
Secant-Tangent Segment ExampleIf O with tangent AC and secant AD, OD = 5 and AB = 6,
calculate AC = ?
A
C
B O 56 D
Mr. Chin-Sung LinERHS Math Geometry
Secant-Tangent Segment ExampleIf 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
A
C
B O 56 D5
4 √6
Mr. Chin-Sung LinERHS Math Geometry
Segments Formed by Two Secants
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Segment TheoremIf 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 OProve: AD · AB = AE · AC
A
CE
B
O
D
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Segment TheoremIf 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 OProve: AD · AB = AE · AC
A
CE
B
O
D
1
2
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Segment TheoremGiven: 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. 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)
A
CE
B
O
D
1
2
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Segment ExampleIf O with secants AC and AE, OC = DE = x, AD = 10 and AB =
8, calculate BC = ?
ACB O
10
8
DE
x
x
Mr. Chin-Sung LinERHS Math Geometry
Secant-Secant Segment ExampleIf 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
BC = 12
ACB O
10
8
DE
6
12
Mr. Chin-Sung LinERHS Math Geometry
Exercises
Mr. Chin-Sung LinERHS Math Geometry
Exercise 1 O has a tangent AF and two secants AC and AB. If AD = 3, CD = 9, and AE =
4, find AF = ? BE = ?
O
B
AD
E
C
3
4
9
F
Mr. Chin-Sung LinERHS Math Geometry
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
O
B
AD
6
E
C
3
4
9
F
5
Mr. Chin-Sung LinERHS Math Geometry
Exercise 2
C has two chords AF and DE. If AP = 6 and PF = 2, EP = 3, and CM = 3, then CN = ?
C
A
DE
F
2
6
3M
N
3
P
Mr. Chin-Sung LinERHS Math Geometry
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 = (52 - 3.52)1/2
C
A
DE
F
2
6
34
M
N
3
P
5
5
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate Plane
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate PlaneA 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?
O
B (0, 5)
5
D (0, –5)
x
C (-5, 0)x
y
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (5, 0)y
x
Circles in a Coordinate PlaneA 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 O
B (0, 5)
5
D (0, –5)
x
C (-5, 0)x
y
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (5, 0)y
x
Circles in a Coordinate PlaneA 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?
O
B (0, r)
r
D (0, –r)
x
C (-r, 0)x
y
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (r, 0)y
x
Circles in a Coordinate PlaneA 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
O
B (0, r)
r
D (0, –r)
x
C (-r, 0)x
y
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (r, 0)y
x
Circles in a Coordinate PlaneA 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?
(2, 4)
B (2, 9)
5
D (2, –1)
x
C (-3, 4)
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (7, 4)|y -4|
|x – 2|
y = 4
x = 2
Circles in a Coordinate PlaneA 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 (2, 4)
B (2, 9)
5
D (2, –1)
x
C (-3, 4)
y = 4
x = 2
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (7, 4)|y -4|
|x – 2|
Circles in a Coordinate PlaneA 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?
(h, k)
B (h, k+r)
r
D (h, k–r)
x
C (h-r, k)
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (h+r, k)|y -k|
|x–h|
y = k
x = h
Circles in a Coordinate PlaneA 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
(h, k)
B (h, k+r)
r
D (h, k–r)
x
C (h-r, k)
y = k
x = h
y P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
A (h+r, k)|y -k|
|x–h|
Equation of a CircleCenter-radius equation of a circle with radius r and
center (h, k) is
(x – h)2 + (y – k)2 = r2
(h, k)
r
P (x, y)
Mr. Chin-Sung LinERHS Math Geometry
Center of a CircleA 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?
C (h, k)r
P (x1, y1)
Mr. Chin-Sung LinERHS Math Geometry
Q (x2, y2)
r
Center of a CircleA 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) = ( , )
r
P (x1, y1)
Mr. Chin-Sung LinERHS Math Geometry
Q (x2, y2)
r
x1 + x2 y1 + y2 2 2
C (h, k)
Center of a Circle ExampleA 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)
r
P (5, 7)
Mr. Chin-Sung LinERHS Math Geometry
Q (-1, -1)
r
C (h, k)
Center of a Circle ExampleA 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)
r
P (5, 7)
Mr. Chin-Sung LinERHS Math Geometry
Q (-1, -1)
r
5 + (-1) 7 + (-1) 2 2 C (h, k)
Radius of a CircleA circle has a diameter PQ with end-points at P (x1, y1) and
Q (x2, y2). What is the radius (r) of the circle?
C (h, k)r
P (x1, y1)
Mr. Chin-Sung LinERHS Math Geometry
Q (x2, y2)
r
Radius of a CircleA 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
C (h, k)r
P (x1, y1)
Mr. Chin-Sung LinERHS Math Geometry
Q (x2, y2)
r
Radius of a Circle ExampleA circle has a diameter PQ with end-points at P (5, 7) and Q
(-1, -1). What is the radius (r) of the circle?
Mr. Chin-Sung LinERHS Math Geometry
r
P (5, 7)
Q (-1, -1)
r
C (h, k)
Radius of a Circle ExampleA 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
Mr. Chin-Sung LinERHS Math Geometry
r
P (5, 7)
Q (-1, -1)
r
C (h, k)
Circles in a Coordinate Plane Exercise
(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?
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate Plane Exercise
(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)
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate Plane Exercise
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?
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate Plane Exercise
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)
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate Plane Exercise
Based on the diagram,
(a) write an equation of the circle
(b) Find the area of the circle
Mr. Chin-Sung LinERHS Math Geometry
Circles in a Coordinate Plane Exercise
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
Mr. Chin-Sung LinERHS Math Geometry
Q & A
Mr. Chin-Sung LinERHS Math Geometry
The End
Mr. Chin-Sung LinERHS Math Geometry