a radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l c t line l...
DESCRIPTION
Two tangent segments drawn to a circle from the same exterior point are congruent. A C B O CA and CB are tangent segments. CA CBTRANSCRIPT
![Page 1: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/1.jpg)
![Page 2: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/2.jpg)
A radius drawn to a tangent at the point of tangency is
perpendicular to the tangent.
l
C
T
Line l is tangent to Circle
C at point T.
CT l at T
![Page 3: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/3.jpg)
Two tangent segments drawn to a circle from the same
exterior point are congruent.A
C
B
O
CA and CB are
tangent segments
.
CA CB
![Page 4: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/4.jpg)
1 2
3
4
5
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O
A
B
CENTRALANGLE
--VERTEX IS AT THE CENTER--SIDES ARE
RADII--ITS MEASURE IS
EQUAL TO THE MEASURE OF ITS
INTERCEPTED ARC
1
m AOB mAB
![Page 6: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/6.jpg)
O
A
B
1
The arc of an
angle is the
portion of the circle
in the INTERIOR
of the angle.
![Page 7: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/7.jpg)
O
A
BC
INSCRIBEDANGLE
--VERTEX IS ON THE CIRCLE--SIDES ARE CHORDS
--ITS MEASURE IS EQUAL TO HALF THE
MEASURE OF ITS INTERCEPTED ARC
12
m ACB mAB
![Page 8: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/8.jpg)
ANGLE FORMED BY
CHORD(SECANT) AND TANGENT
WITH VERTEX ON CIRCLE
--ITS MEASURE IS EQUAL TO HALF
THE MEASURE OF ITS INTERCEPTED
ARC
C 3O
A
B12
m CAB mAB
![Page 9: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/9.jpg)
ANGLE FORMED BY TWO CHORDS
(SECANTS) WHOSE VERTEX IS IN THE INTERIOR OF THE
CIRCLE, BUT NOT AT THE CENTER
--ITS MEASURE IS EQUAL TO HALF THE SUM OF THE MEASURES OF ITS INTERCEPTED
ARCS (ITS ARC AND THE ARC OF ITS VERTICAL ANGLE)
C
A
B
D
E
1( )2
m CAB mCB DE
![Page 10: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/10.jpg)
AN ANGLE WHOSE VERTEX
IS IN THE EXTERIOR
OF A CIRCLE MAY BE FORMED
BY:
A TANGENT AND A
SECANT
TWO SECANTS
ORTWO
TANGENTS
B
ACD
BA
C
D
A
B
C
D
E
![Page 11: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/11.jpg)
IN EACH CASE, THE MEASURE OF
THE ANGLE IS ONE-HALF THE DIFFERENCE OF
THE MEASURES OF THE INTERCEPTED
ARCS.
![Page 12: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/12.jpg)
B
ACD
A
B
C
D
BA
C
DE
1( )2
m A mBCD mBD
1( )2
m A mCD mBE
1( )2
m A mBC mBD
![Page 13: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/13.jpg)
![Page 14: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/14.jpg)
Two points on a circle will always
determine:
![Page 15: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/15.jpg)
A CHORD
![Page 16: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/16.jpg)
TWO RADII
![Page 17: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/17.jpg)
A MINOR ARC
![Page 18: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/18.jpg)
A MAJOR ARC
![Page 19: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/19.jpg)
OR TWO SEMICIRCLES
![Page 20: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/20.jpg)
CONGRUENT CENTRAL ANGLES WILL HAVE
CONGRUENT CHORDS AND ARCS (AND VICE-
VERSA) IF AND ONLY IF THEY ARE IN THE SAME
OR IN CONGRUENT CIRCLES!!
![Page 21: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/21.jpg)
IN THE SAME CIRCLE OR IN CONGRUENT
CIRCLES, CONGRUENT CHORDS ARE EQUIDISTANT
FROM THE CENTER.
![Page 22: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/22.jpg)
IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF TWO
CHORDS ARE EQUIDISTANT FROM THE CENTER, THEN
THEY ARE CONGRUENT.
![Page 23: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/23.jpg)
IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF TWO
CHORDS ARE UNEQUAL, THEN THE LONGER CHORD IS
CLOSER TO THE CENTER.
![Page 24: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/24.jpg)
IN THE SAME CIRCLE OR IN CONGRUENT
CIRCLES, IF THE DISTANCES FROM THE
CENTER OF TWO CHORDS ARE UNEQUAL,
THEN THE LONGER CHORD IS CLOSER TO
THE CENTER.
![Page 25: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/25.jpg)
O
“Anything” from the center of a circle
(segment,
radius,
diameter)
A B
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O
perpendicular to a chord
A B
![Page 27: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/27.jpg)
O
A B
bisects “everything” it touches:
![Page 28: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/28.jpg)
O
A B
the chord
C
![Page 29: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/29.jpg)
O
A B
the central angle
C
![Page 30: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/30.jpg)
O
A B
the minor arc
C
![Page 31: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/31.jpg)
O
A B
the major arc
C
![Page 32: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/32.jpg)
Describe how the center,O, can be located.
.AB and CD are chords of O
BA
C
D
![Page 33: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/33.jpg)
Construct the perpendicular bisectors of
the chords. They will intersect at the center!B
AC
D
O
![Page 34: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/34.jpg)
A
B
12
IF TWO INSCRIBED
ANGLES INTERCEPT THE SAME ARC, THEN THEY ARE
CONGRUENT.
![Page 35: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/35.jpg)
B
OA C
D
IF AN ANGLE IS
INSCRIBED IN A
SEMICIRCLE,THEN IT IS
A RIGHT ANGLE.
![Page 36: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/36.jpg)
A
BC
D
O
IF A QUADRILATERAL
IS INSCRIBED IN A CIRCLE, THEN ANY PAIR OF OPPOSITE
ANGLES ARE SUPPLEMENTARY.
![Page 37: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/37.jpg)
abc
da
bc
b
a
c
d
ab cd
2 a bc
ab dc
![Page 38: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/38.jpg)
IF TWO CHORDS OF A CIRCLE INTERSECT, THEN THE PRODUCT OF THE LENGTHS OF THE SEGMENTS ON
ONE CHORD EQUALS THE PRODUCT OF THE LENGTHS OF THE SEGMENTS ON
THE OTHER CHORD. a
bc
dab = cd
![Page 39: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/39.jpg)
IF A TANGENT SEGMENT AND A SECANT SEGMENT INTERSECT IN THE EXTERIOR OF A CIRCLE, THEN THE SQUARE OF THE LENGTH
OF THE TANGENT SEGMENT IS EQUAL TO THE PRODUCT OF THE LENGTHS OF THE
SECANT SEGMENT AND ITS EXTERNAL PART.
a
bc
a2 = bc
![Page 40: A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT l at T](https://reader036.vdocuments.us/reader036/viewer/2022062401/5a4d1b2c7f8b9ab05999975d/html5/thumbnails/40.jpg)
IF TWO SECANT SEGMENTS INTERSECT IN THE EXTERIOR OF A CIRCLE, THEN THE
PRODUCT OF THE LENGTHS OF ONE SECANT SEGMENT AND ITS EXTERNAL
PART IS EQUAL TO THE PRODUCT OF THE LENGTHS OF THE OTHER SECANT
SEGMENT AND ITS EXTERNAL PART.
a
b
dc
ab = cd