circles 2
DESCRIPTION
CIRCLES 2. Moody Mathematics. ANGLE PROPERTIES:. Let’s review the methods for finding the arcs and the different kinds of angles found in circles. Moody Mathematics. The measure of a minor arc is the same as…. …the measure of its central angle. Moody Mathematics. Example:. - PowerPoint PPT PresentationTRANSCRIPT
CIRCLES 2CIRCLES 2
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ANGLE ANGLE PROPERTIES:PROPERTIES:
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Let’s review the Let’s review the methods for finding methods for finding the arcs and the the arcs and the different kinds of different kinds of angles found in angles found in circles.circles.
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The measure of a minor arc is the same as…
…the measure of its central angle.
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75 75
Example:
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The measure of an inscribed angle is…
…half the measure of its intercepted angle.
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88
44
Example:
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The measure of an angle formed by a tangent and secant is …
…half the measure of its intercepted arc.
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Example:
230115
130
65
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The measure of one of the vertical angles formed by 2 intersecting chords
...is half the sum of the two intercepted arcs.
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Example:110
60
85
1(110 60 )2
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The measure of an angle formed by 2 secants intersecting outside of a circle is…
…half the difference of the measures of its two intercepted arcs.
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Example:
90
20
35 1(90 20 )2
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The measure of an angle formed by 2 tangents intersecting outside of a circle is…
…half the difference of the measures of its two intercepted arcs.
Moody MathematicsMoody Mathematics
Example:
250
110
70 1(250 110 )2
PROPERTIES: PROPERTIES: Complete the Complete the theorem relating theorem relating the objects the objects pictured in each pictured in each frame.frame.
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Note: Note: Many Many of our theorems of our theorems begin the same begin the same way, “In the same way, “In the same circle, circle, or in or in congruent congruent circlescircles…”…”
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So: So: We will We will just start “In the just start “In the same circle*…” same circle*…” where the where the ** represents the represents the rest of the phrase. rest of the phrase.
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All radii in the same circle,* …
...are congruent.
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In the same circle,* Congruent central angles...
...intercept congruent arcs.
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In the same circle,* Congruent Chords...
...intercept congruent arcs.
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Tangent segments from an exterior point to a circle…
...are congruent.
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The radius drawn to a tangent at the point of tangency…
...is perpendicular to the tangent.
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If a diameter (or radius) is perpendicular to a chord, then…
...it bisects the chord……and the arcs.
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In the same circle,* Congruent Chords...
...are equidistant from the center.
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Example: Given a circle of radius 5” and two 8” chords. Find their distance to the center.
5
4
4
2 2 24 5x 3x
x
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If two Inscribed angles intercept the same arc...
...then they are congruent.
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If an inscribed angle intercepts or is inscribed in a semicircle …
...then it is a right angle.
180
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If a quadrilateral is inscribed in a circle then each pair of opposite angles …
...must be supplementary.
(total 180o)
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If 2 chords intersect in a circle, the lengths of segments formed have the following relationship:
a b c d
a
b
c d
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3 4 2 c 3
4
c
Example:
2
6c
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If 2 secants intersect outside of a circle, their lengths are related by…
a b c d
a
c
bd
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8 3 2c
c
32
Example:
8
12c
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If a secant and tangent intersect outside of a circle, their lengths are related by…
a a c d
a
c
d
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4 (4 5)a a a
5
4
Example:
6a
Let’s Let’s Practice!Practice!
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Example: Given
50
P
A
B
C
D
P
mAB
mBC
mABC
mADB
mACD
50
130
180
310
230
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Example:
200100
160
80
x
y
z
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55 55
Example:
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Example:
80
30
25 1(80 30 )2
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Example: Given a circle of radius 13” and two 24” chords. Find their distance to the center.
1312
2 2 212 13x 5x
x12
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55
35
70
x180
y
z
Example:
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5 (15 5)x x x
15
5
Example:
10x
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Example:110
40
75
1(110 40 )2
x
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Example:
230
130
50 1(230 130 )2
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140
x
160
y40
120
Example:
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8 12 6x
x
12 68
9x
Example:
Example: Of the following quadrilaterals, which can not always be inscribed in a circle?
A.Rectangle
B.Rhombus
C.Square
D.Isosceles Trapezoid
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90
50
x
y
z
25x
45y
70z
Example:
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Example: 160
xy
z
80x
100y
50z
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Example: Regular Hexagon ABCDEF is inscribed in a circle. A B
C
DE
F
mACE 240
THE END!THE END!Now go practice!