Download Geometry 10-1 Circles and Circumference

Geometry 10-1 Circles and Circumference
A. Parts of Circles
1. A circle is the locus of all points equidistant from a given point called the
center of the circle.
2. A circle is usually named by its ________ point.
3. The circle below is called circle K, written ○K
C
A
Any segment with endpoints that
are on the circle is a ___________
of the circle. AB and CD are
chords.
K
B
F
A chord that passes through the
______________ of the circle is a
diameter of the circle. CD is a
diameter of the circle.
Ex 1: Identify parts of the circle.
a.) Name the circle.
b.) Name a radius of the circle.
c.) Name a chord of the circle.
d.) Name a diameter of the circle.
D
Any segment with endpoints that
are _________________ and a
point on the circle is a radius. KF,
KC, and KD are radii of the circle
4. The distance from the center of the circle to any point on the circle is always
the same, therefore, all radii are congruent.
5. A diameter is composed of two radii, so, d = 2r
Ex 2: Circle R has diameters ST and QM.
a.) If ST = 18, find RS.
b.) If RM = 24, find QM.
c.) If RN = 2, find RP.
Ex 3: The diameters of ○X, ○Y, and ○Z are 22 mm, 16 mm, and 10 mm
respectively.
a.) Find EZ.
b.) Find XF.
B. Circumference.
1. The circumference is the distance around a circle.
2. The circumference around a circle is often represented by the letter C.
3. The ratio between the circumference of a circle and it’s diameter is about 3:1.
For each diameter, circumference is a little more than 3.
4. For a circumference of C units, diameter of d, or radius of r,
Ex 4: Find the missing value for each circle.
a.) Find C if r = 13 inches.
3√2
b.) Find C if d = 6 millimeters.
K
c.) Find d and r to the nearest tenth if C = 13Π
Ex 5: Find the exact circumference for ○K.
Geometry 10-2 Angles and Arcs
A. Angles and Arcs.
1. A central angle has the __________ of the circle as the vertex and its sides
contain two radii of the circle.
2. Sum of Central Angles -the sum of the
measures of a circle with no interior points
in common is 360.
1
2
3
Ex 1: Refer to ○T.
a.) Find m<RTS.
b.) Find m<QTR
(8x – 4)o
(13x – 3)o
Minor Arc
Major Arc
Semicircle
by the two letters of the
endpoints
by the letters of the two
endpoints and another
point on the arc.
Greater than 180, but
less than 360
mDFE = 360 – mDE
mDFE = _________
by the letters of the two
endpoints and another
point on the arc.
mJKL = ________
Example
Named:
Arc degree The measure of the central
measure = angel is less than 180.
m<ABC = 110o so mAC =
110
mJML = ________
3. Theorem 10-1 -In the same or in congruent circles, two arcs are congruent if
and only if their corresponding central angles are congruent.
4. Postulate 10-1 Arc Addition Postulate -The measure of an arc formed by two
adjacent arcs is the sum of the measures of the two arcs.
Ex 3: In ○P, m<NPM = 46 , PL bisects <KPM , and OP ┴ KN Find each measure.
a.) mOK
b.) mLM
c.) mJKO
Ex 4: This graph shows the percent of each type of bicycle sold in the United
States in 2001.
a.) Find the measurement of the central
angle representing each category.
Youth - 26%(360) = _____
Bicycles Bought In 2001 (By Type)
Mountain - ____________ = ____
ComfortRoadHybrid-
Mountain
37%
Youth
26%
b.) Is the arc for the wedge named
Youth congruent to the arc for the
combined wedges named Road and
Comfort?
Comfort
21%
9%
Hybrid
7%
Road
B. Arc Length
1. An arc is part of a circle, so arc length is part of the circumference.
s = Π r (degree measure / 180) or (degree measure) = s
360
2Πr
Ex 5: In ○B, AC = 9, and m<ABD = 40. Find the length of AD
Geometry 10-3 Arcs and Chords
A. Arcs and Chords
1. Theorem 10-2 -In a circle or in congruent circles, two minor arcs are congruent
if and only if their corresponding chords are congruent.
Abb. -In ○, 2 minor arcs are , corr. chords are .
B
-In ○, 2 chords are , corr. minor arcs are  .
A
D
C
Ex: If AB  CD then AB  CD, and if AB  CD then AB  CD
Ex 1: Finish the proof of Theorem 10-2
Given: ○X, UV  YW
Prove: UV  YW
U
V
X
W
Statements
Reasons
1.) ○X, UV  YW
1.) Given
2.) <UXV  <WXY
________________________
2.) If arcs are
Y
________________________________
3.) UX  VX  WX  YX
_________________________________
3.)
4.) _____________________________
_________________________________
4.)
5.) UV  YW
_________________________________
5.)
B. Diameters and Chords
1. Theorem 10-3: In a circle, if a diameter (or radius) is perpendicular to a chord,
then it bisects the chord and its arc.
Ex: If BA ┴ TV, then _______ and _________  ___________
B
C
T
V
A
Ex 2: Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK,
which is 16 cm. long.
a.) If mHL = 53, find mMK.
M
W
b.) Find JL.
J
H
K
L
2. Theorem 10-4 In a circle or in congruent circles, two chords are congruent if
and only if they are equidistant from the center.
Ex 3: Chords EF and GH are equidistant from the center. If the radius of d P is 15
and
EF = 24, find PR and RH.
E
Q
F
P
H
R
G
Geometry 10-4 Inscribed Angles
A. Inscribed angles
1. An inscribed angle is an angle that has its vertex and its sides contained in the
chords of the circle.
B
Vertex B is on
the circle.
A
AC is the arc
intercepted by
<ABC.
C
AB and BC are
chords on the
circle.
2. Theorem 10-5 -___________________________________________ -If an
angle is inscribed in a circle, then the measure of the angle equals one-half the
measure of its intercepted arc (or the measure of the intercepted arc is twice the
measure of the inscribed angle).
Ex: m<ABC =(mAC ) or 2( m<ABC ) = mAC
Ex 1: In ○F, mWX = 20, mXY = 40, mUZ = 108, and mUW = mYZ. Find the
measures of the numbered angles.
3. Theorem 10-6 -If two inscribed angles of a circle (or congruent circles)
intercept congruent arcs or the same arc, then the angles are congruent.
Abbreviations:
Inscribed <’s of  arcs are .
Inscribed <’s of same arc are .
<DAC  <__________
Ex 2:
Given: ○C with QR  GF, and JK  HG
Prove: ▲PJK  ▲EHG
Statements
Reasons
1.) QR  GF, JK  HG
2.) QR  GF
1.) ________________________
2.)
3.) <GEF intercepts FG
<QPR intercepts QR
4.)
___________________________
3.)
___________________________
4.)
__________________________
5.)
___________________________
5.)
__________________________
6.) ▲PJK  <EHG
___________________________
6.)
B. Angles of Inscribed Polygons
1. Theorem 10-7 -If an inscribed angle intercepts a semicircle, the angle is a right
angle.
ADC is a semicircle so m<ABC = 90o
Ex 3: Triangles TVU and TSU are inscribed in ○P with VU  SU. Find the
measure of each numbered angle if m<2 = x + 9 and m<4 = 2x + 6.
Ex 4: Quadrilateral QRST is inscribed in ○M.
If m<Q = 87 and m<R = 102. Find m<S and m<T.
2. Theorem 10-8 -If a quadrilateral is inscribed in a circle, then its opposite
angles are supplementary.
Geometry 10-5 Tangents
A. Tangents
1. BC is _________________ to ○A , because the line containing BC intersects
the circle in exactly one point.
2. This point is called the _____________________.
3. Theorem 10-9 -If a line is tangent to a circle, then it is perpendicular to the
radius drawn to the point of tangency.
If AB is a tangent, AB ┴ AK
Ex 1: RS is tangent to d Q at point R. Find the diameter.
4. Theorem 10-10 -In a plane, if a line is ____________________________ to a
radius of a circle at the endpoint on the circle, then the line is a tangent to the
circle.
Ex 2: a.) Determine whether
BC is tangent to ○A
b.) Determine whether
WE is tangent to ○D.
5. Theorem 10-11 -If two segments from the same exterior point are tangent to a
circle, they are congruent.
AB  BC
Ex 3: Find x, assume that segments that appear to be tangent are.
B. Circumscribed Polygons
1. Polygons can be circumscribed about a circle, or the circle is inscribed in the
polygon. (every side of the polygon is tangent to the circle.)
Ex 4: Find the perimeter of the triangle. Given: AD = CB
Geometry 10-6 Secants, Tangents, and Angle Measures
A. Intersections on or inside a circle.
1. A line that intersects a circle in exactly two points is called a ______________.
-A secant of a circle contains a chord of the circle.
2. Theorem 10-12 -If two secants intersect in the interior of a circle, then the
measure of an angle formed is one-half the sum of the measures of the arcs
intercepted by the angle and its vertical angle.
Examples:
m<1 = (mAC + mDB)
2
m<2 = ( mAD + mBC)
2
Ex 1: Find m<4 if mFG = 88 and mEH = 76
3. Theorem 10-13 -If a secant and a
tangent intersect at the point of tangency,
then the measure of each angle formed is
____________ the measure of its intercepted arc.
Ex 2: Find m<RPS if mPT = 114o and mTS = 136o
B. Intersections outside of a circle.
1. Theorem 10 -14 -If two secants, a secant and a tangent, or two tangents
intersect in the exterior of a circle, then the measure of the angle formed is onehalf the positive difference of the measures of the intercepted arcs.
Two secants
Tangents
Secant -Tangent
Two
m<A = ½ (mDE – mBC)
m<A = ½ (mDC – mBC)
m<A = ½
(mBDC- mBC)
Ex 3: Find x.
Ex.5: Find x
Ex.4: A jeweler wants to craft a
pendant with the shape shown. Use
the figure to determine the measure
of the arc at the bottom of the
pendant
Geometry 10-7 Special Segments in a Circle
A. Segments intersecting in a circle.
1. Theorem 10-15 -If two chords intersect in a circle, then the products of the
measures of the segments of the chords are equal.
So AE * EC = DE * EB
Ex 1: Find x.
Ex 2: Biologists often examine organisms under microscopes. The circle
represents the field of view under the microscope with a diameter of 2 mm.
Determine the length of the organism is it is located 0.25 mm from the bottom of
the field of view. Round to the nearest hundredth of a millimeter.
0.25 mm
B. Segments intersecting outside a Circle
1. Theorem 10-16 -If two secant segments are drawn to a circle from an
___________ _________, the product of the measures of one secant segment and
its external secant segment is equal the product of the measures of the other secant
segment and its external secant segment.
AB * AC = AD * AE
Ex 3: Find HJ if EF = 10, EH = 8, and FG = 24.
2. If a _______________ segment and a _______________ segment are drawn to
a circle from an exterior point, then the square of the measure of the tangent
segment is equal to the product of the measures of the secant segment and its
external secant segment.
(WX )2 = WZ * WY
Ex 4: Find x, assume that segments that appear to be tangent are.
Ex 5: Find x, assume that segments that appear to be tangent are.
Geometry 10-8 Equations of Circle
A. Equation of a Circle
1. Suppose the center of a circle is at (3, 2), and the radius is 4.
2. Use the distance formula to determine the distance to a point on the circle.
this is the general form for an equation of a circle.
3. An equation for a circle with center at (h, k) and radius of r units, is
Ex 1: Write an equation for each circle.
a.) center at (4, -3), r = 6
b.) center at (-12, -1), d = 16.
Ex 2: A circle with a diameter of 10 has its center in the first quadrant. The lines y
= -3 and x = -1 are tangent to the circle. Write an equation of the circle.
Ex 3: a.) Graph (x - 2)2 +( y + 3)2 = 4
b.) Graph (x – 3)2 + y2 = 16
c) Graph x2 + (y + 2)2 = 9