Download Geometry Chapter 13 - Eleanor Roosevelt High School

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: mACB = (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: mACB = (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: mACB = (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: mACB = (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: 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
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: mACB = 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: mACB = 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: mB + mD= 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: mB + mD= 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) mABC = (1/2) m AB
O
2) mABD = (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) mABC = (1/2) m AB
1
O
2) mABD = (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. mABC + m2 = 90
A
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
C
10. (1/2) mAOB = mABC
11. mABC = (1/2) m AB
12. 180 - mABC = (1/2) (360 - m AB)
13. mABD = (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: mA = (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: mA = (1/2) (m BEC - m BC)
B
1. Draw BC, form 1 and 2
1
2. AB = AC
3. m1 = m2 = (1/2) m BC
O
A
4. mA + m1 + m2 = 180
2
5. mA + m BC = 180
E
6. m BC + m BEC = 360
C
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 )
ERHS Math Geometry
Mr. Chin-Sung Lin
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 + 90 + 90 + mBOC = 360
5. mA + mBOC = 180
A
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 )
ERHS Math Geometry
B
1
O
2
E
C
Mr. Chin-Sung Lin
Given: O with tangents AB and AC
Prove: mA = (1/2) (m BEC - m BC)
1. Draw BC, form 2
2
2. Extend AC, form 1
3. m2 = (1/2) m BC
B
O
A
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 )
ERHS Math Geometry
1
E
C
D
Mr. Chin-Sung Lin
Given: O with secants AD and AE
Prove: mA = (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: mA = (1/2) (m DE - m BC)
1. Draw DC
B
1
O
A
2. m2 = mA + m1
3. mA = m2 - m1
2
C
E
4. m2 = (1/2) m DE, m1 = (1/2) m BC
5. mA = (1/2) (m DE - m BC)
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: O with secants AB and AC
Prove: mA = (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. m3 = m4, m7 = m8
C
4. m5 = 180 - 2 m3, m9 = 180 - 2 m7
O
9
7
8
E
5. m3 = mBOA + mBAO
6. m7 = mCOA + mCAO
7. m5 + m9 = 180 - 2 m3 + 180 - 2 m7
= 360 - 2(m3 + m7)
ERHS Math Geometry
Mr. Chin-Sung Lin
Given: O with secants AD and AE
Prove: mA = (1/2) (m DE - m BC)
8. m5 + m9 = 360 - 2(mBOA + mBAO + mCOA +
mCAO) = 360 - 2(mBOC + mA)
9. m5 + m9 + 2(mBOC + mA) = 360
D
10. m5 + m9 + mBOC + mDOE = 360
B
11. mBOC - mDOE + 2mA = 0
12. 2 mA = mDOE - mBOC
A
13. mA = (1/2) (mDOE- mBOC)
14. mA = (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: mA = (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: mA = (1/2) (m DEC - m BC)
B
1. Draw BC
2. m2 = mA + m1
2
O
A
3. mA = m2 - m1
4. m2 = (1/2) m DEC, m1 = (1/2) m BC
1
E
C
5. mA = (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 mG
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 mG
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: mAMC = mBMD
= (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: mAMC = mBMD = (1/2) (m AC + m BD)
1. Draw BC
2.
3.
4.
5.
6.
7.
A
mAMC = mBMD
mAMC = m1 + m2
m1 = (1/2) m AC
m2 = (1/2) m BD
C
mAMC = (1/2) m AC + (1/2) m BD
mAMC = mBMD = (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
= ? mA = ?
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
= ? mA = ?
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 mEAB = 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 mEAB = 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. m1 = m2 (Congruent inscribed angles)
4
3. m3 = m4 (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. m1 = (1/2) m BC (Tangent-chord angles theorem)C
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)
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. 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)
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