11/7/2016
Circles - angles formed by radii, chords, tangents, secants
Formulas for Angles in Circles
Formed by Radii, Chords, Tangents,
Secants
Formulas for Working with Angles in Circles
{Intercepted arcs are arcs "cut off or "lying between" the sides of the specified angles.)
There are basically five circle formulas that
you need to remember:
1.
Central Angle:
A central angle is an angle formed by two intersecting
radii such that its vertex is at the center of the circle.
Central Angle = Intercepted Arc
m<A0B =mAB
<AOB is a central angle.
Its intercepted arc is the minor arc from A to B.
m<AOB = 80°
Theorem involving central angles'.
In a circle, or congruent circles, congruent central angles have congruent arcs.
2.
Inscribed Angle:
An inscribed angle is an angle with its vertex "on" the
circle, formed by two intersecting chords.
Inscribed Angle = 1 Intercepted Arc
2
m<ABC = — rnAC
2
<ABC is an inscribed angle.
Its intercepted arc is the minor arc from A to C.
m<ABC=5Q°
Vnz>r»iz>>7 ci/iizih'nnr íuon/nÍMa
httD://www.reaentSDren nra/reaAnts/mafh/nAnmptrv/nnl5/rJrr.foAnalpR htm
A auadrilateral inscribed, in a circle is called, a
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11/7/2016
Circles - angles formed by radii, chords, tangents, secants
uptviui o»&«««»tc/rtj uiruiviiig iitouwcu
angles'.
An angle inscribed in a
semi-circle is a right angle.
m<ABC’=l(mAC) = 1(180°) = 90°
cyclic quadrilateral.
The opposite angles in a cyclic
quadrilateral are
supplementary.
mo:=1 (ynScB] = - (mSABj
Mt+miy = ■l^m'DCS+mZWS)
In a circle, inscribed circles that intercept
the same arc are congruent.
n!<r+m<y =^1(360°) = 180°
3. Tangent Chord Angle:
An angle formed by an intersecting tangent and chord has
its vertex "on” the circle.
B
Tangent Chord Angle =
1 Intercepted Arc
2
m4A.BC = — rnAB
2
<ABC is an angle formed by a tangent and chord.
Its intercepted arc is the minor arc from A to B.
m<ABC = 60°
4. Angle Formed Inside of a Circle
by Two Intersecting Chords:
When two chords intersect "inside" a circle, four angles
are formed. At the point of intersection, two sets of
vertical angles can be seen in the comers of the X that is
formed on the picture. Remember: vertical angles are
equal.
B
Angle Formed Inside by Two Chords =
Isum of Intercepted Arcs
2
m<BED = An AC + mBDr
Once you have found ONE of these angles, you
automatically know the sizes of the other three
by using your knowledge of vertical angles
(being congruent) and adjacent angles forming a
straight line (measures adding to 180).
http:/A/vww. regentsprep.org/regents/math/geometry/gp15/circleangles.htm
<BED is formed by two intersecting chords.
Its intercepted arcs are BD and CA .
[Note: the intercepted arcs belong to the set
of vertical angles.]
m<BED=j(70 + 170) = 1^240) = 120°
also, m<CEA = 120° (vetical angle)
m<BEC and m<DEA = 60° by straight line.
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Segment Rules in Circles: Chords, Secants, Tangents
Rules for Dealing with Chords,
Secants, Tangents in Circles
Topic Index | Geometry Index | Regents Exam Prep Center .,
Theorem 1:
If two chords intersect in a circle, the product of the lengths of the
segments of one chord equal the product of the segments of the other.
Intersecting Chords Rule:
(segment piece)x(segment piece) =
(segment piece)*(segment piece)
a*b= c*d
Theorem Proof:
D
Given: Chords AB and CD
B Prove: AE*EB = CE*ED
Statements
Reasons
1.
Chords A45 and CD
1. Given
2.
Draw ~ÃC, BD
2. Two points determine only one line.
3.
Ol <C; <B <D^
3. If two inscribed angles intercept the
same arc, the angles are congruent.
4.
MDE-ACBE
4.
5.
A5 ED
5. Corresponding sides of
triangles are in proportion.
CE-EB
6.
AEEB = CEED
http://www.regentsDrep.org/regents/math/geometry/gDl4/circlesegments.htm
6.
AA - If two angles of one triangle
are congruent to the corresponding
angles of another triangle, the
triangles are similar.
similar
In a proportion, the product of the
means equals the product of the
extremes.
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11/7/2016
Segment Rules in Circles: Chords, Secants, Tangents
Theorem 2:1 Ifwo secant segments are drawn to a circle from the same external point,
-——1 ----- 1 the product of the length of one secant segment and its external part is
I
equal to the product of the length of the other secant segment and its
external part.
a *b ~ c *â
Theorem 3:
Secant-Secant Rule:
(whole secant)x(external part) =
(whole secant)x(external part)
If a secant segment and tangent segment are drawn to a circle from the
same external point, the product of the length of the secant segment and its
external part equals the square of the length of the tangent segment.
Secant-Tangent Rule:
(whole secant)x(external part) =
(tangent)2
This theorem can also be stated as ’’the tangent being the mean proportional between the whole secant and its
external part" (which yields the same final rule:
whfiifi8fiGant ----- —tai^gent------------(wt0|e
secairt)*(e}denialpart)=(tàiigeiit)?
tangent external fart
Topic Index I Geometry Index I Regents Exam Prep Center
Created by Donna Roberts
Copyright 1998-2012 http://regentsprep.org
Oswego City School District Regents Exam Prep Center
http://w w w. regentsprep. or g/r egents/m ath/geom etry/gp14/ci rclesegm ents. htm
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Circles - angles formed by radii, chords, tangents, secants
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5. Angle Formed Outside of a Circle by the Intersection of:
"Two Tangents" or "Two Secants" or "a Tangent and a Secant".
Two Secants:
<ACE is formed by two secants
intersecting outside of circle O.
The intercepted arcs are minor arcs BD and
m<ACE=l(80-20) = 30°
a Tangent and a Secant:
<ABD is formed by a tangent and a secant
intersecting outside of circle O.
The intercepted arcs are minor arcs J1C and
•
m<ABD=l( 100-30) = 35°
m<ABD = —(mAD-mAC
2‘
'
AP-AWW-regentsprep
.org/regentS/math/geometry/qp15/circleanqles.htm
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TANGENTS, SECANTS, AND CHORDS
#19
The figure at right shows a circle with three lines lying on
a flat surface. Line a does not intersect the circle at all.
Line b intersects the circle in two points and is called a
SECANT. Line c intersects the circle in only one point
and is called a TANGENT to the circle.
-------- > a
TANGENT/RADIUS THEOREMS:
1.
Any tangent of a circle is perpendicular to a radius of the
circle at their point of intersection.
2.
Any pair of tangents drawn at the endpoints of a diameter
are parallel to each other.
A CHORD of a circle is a line segment with its endpoints on the circle.
DIAMETER/CHORD THEOREMS:
1. If a diameter bisects a chord, then it is perpendicular to the chord.
2.
If a diameter is perpendicular to a chord, then it bisects the chord.
ANGLE-CHORD-SECANT THEOREMS:
mZl = (mAD + mBC)
AE • EC = DE • EB
mZP = l(mRT-mQS)
PQPR = PSPT •
Example 1
Example 2
If the radius of the circle is 5 units and
AC =13 units, find AD and AB.
In OB, EC = 8 and AB = 5. Find BF.
AD±CD and AB±CD by Tangent/Radius
The diameter is perpendicular to the chord,
therefore it bisects the chord, so EF = 4. AB
is a radius and AB =5. EB is a radius, so
EB = 5. Use the Pythagorean Theorem to
find BF: BF2 + 42 = 52, BF = 3.
Theorem, so (AD)2 +(CD)2 = (AC)2 or
(AD)2 +(5)2 =(13)2. So AD = 12 and
AB-AD so AB = 12.
GEOMETRY Connections
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