Download Act. 4.3: Angles Formed by Chords, Tangents and Secants

Learning Target:
I can review properties of angles and
segments in circles to determine their
measure and length.
Circles Review: Properties, Angles
and Segments
Review
Agenda:
1. Do Now
2. Embedded Assessment Self-Assess
3. Circles Properties Review
4. Independent Practice
5. Debrief/Note Sheet Creation
DO NOW 3/27:
Find the radius BO if AB = 4in and
AO = 5 in.
Embedded Assessment: Vertigo Round
Definitions:
Radii, Chords, and Tangents
 A radius is a line segment with one endpoint at the center of
the circle and the other endpoint on the circle.
 A chord is a line segment with both endpoints on the circle
 A tangent is a line that touches the circle at one point
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(the “point of tangency”)
D
R
T
O
C
Property 1: Radius and Tangent
 When a radius and tangent meet, it forms a 90˚ angle.
N
R
T
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Property 2: Radius and Chords
 A radius that is perpendicular to a chord bisects that chord.
R
x
x
C
O
D
Two Chords…
 Two congruent chords are always the same distance from the
center.
D
R
x
x
C
O
H
Property 3: Two Tangents…
 Two tangents starting from the same point outside a circle
are congruent to the point of tangency.
T
x
x
N
A
O
Circles Review: Arcs, Central and
Inscribed Angles
Review
Definition: Arcs
 An arc is the section of the circumference of a circle between
two points.
C
A
O
Definition:
Central and Inscribed Angles
 A central angle is an angle with its vertex at the center of the
circle (sides are radii)
 An inscribed angle is an angle with its
C
T
vertex on the circle (sides are chords)
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N
S
I
Properties:
Arcs, Central and Inscribed Angles
 The measure of a central angle is the same as the arc it
intercepts.
 The measure of an inscribed angle is
½ of the arc it intercepts.
75˚
C
T
75˚
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20˚
S
I
40˚
Circles Review: Angles Formed by
Chords, Tangents and Secants
Review
Equation 1:
Angles Formed by Chords
 The angle formed by 2 chords is ½ of the sum of the two
arcs.
 x = ½(a+b)
P
L
O
x
b
M
a
x
Q
Equation 2a:
Angles Formed by Secants
 Secant – a line that intersects the circle at 2 points
 The angle formed by 2 secants is ½ the difference of the two
arcs.
 x = ½(a-b)
P
M
L
x
a
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b
N
Q
Equation 2b: Angles Formed by
Tangent and Secant
 The angle formed by a tangent and secant is ½ the difference of
the two arcs.
 x = ½(a-b)
P
A
x
b
a
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Q
R
Equation 3:
Angles Formed by Tangents
 The angle formed by two tangents is the major arc minus
180.
 x = a – 180
P
L
O
x
Q
a
Learning Target:
I can review how to solve problems
involving segments in circles, arc length,
sector area and equations of a circle.
Circles Review: Segment Lengths in
Circles
Review
Agenda:
1. Do Now
2. Circles Properties Review
3. Jeopardy Review Game
4. Embedded Assessment Debrief
/Note Sheet Creation
DO NOW 3/30:
Solve for x.
Chord Segment Length
 When two chords intersect, the products of the two
segments lengths of each chord are equal.
 LA•AQ = MA•AP
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L
A
M
Q
Secants Segment Length
 The product of the whole secant segment and the external secant
segment of each secant are equal.
P
 LP•LM = LQ•LN
M
O
L
N
Q
 Remember! Whole secant • external secant
Tangent and Secant Segment Lengths
 The product of the whole secant segment and the external
secant is equal to the tangent segment squared.
 AR•AQ = AP2
P
O
A
Q
R
Learning Target:
I can review and practice arc lengths, sector
area and equations of circles to prepare for
the unit exam.
Act. 4.5: Area, Circumference,
Sectors and Arc Lengths
Review
Circumference and Area
 The circumference of a circle is the distance around the
outside of the circle.
 C = 2πr
 The area is the space the circle covers
 A = πr2
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Sector and arc length
 Arc˚/360˚ = fraction of a circle
 A sector is a fraction of the area
 Sector area = (arc˚/360 ˚)(πr2)
 The arc length is a fraction of
the circumference
 Arc length = (arc˚/360 ˚)(2πr)
O
Act. 4.6: Equation of a Circle
Review
Equation of a Circle
 The equation of a circle is made up of 3 parts:
 The radius (r)
 The center point (h,k)
 Another point on the circle (x,y)
 r2 = (x-h)2 + (y-k)2
(h,k)
r
(x,y)