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Circle
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Page 2 ©National Science Foundation
Circle
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Definition
"The set of all points equidistant from the center".
The locus of all points a fixed distance from a given (center) point.
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Terms related to a circleCenter - a point inside the circle from which all points on the circle are equidistant
Radius - the distance from the center to any point on the circle.
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Terms related to the circle
Chord - a line segment joining any two points on a circle.
Diameter - a chord passing through the center.
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Terms related to the circle
•Tangent - A line passing a circle and touching it at just one point.
•Secant - A line that intersects a circle at two points.
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Terms related to a circle
ARC – a subset of a circle
Minor Arc < 180 Major Arc > 180Semicircle = 180
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Terms related to a circle
Central Angle – angle made by two radii
Inscribed Angle – angle made by two chords intersecting ON a circle
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Terms related to a circle
•Circumference - the distance around the circle. Strictly speaking a circle is a line, and so has no area.
•What is usually meant is the area of the region enclosed by the circle.
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1 Kings 7:23-24 (New American Standard Bible)
23Now he made the sea of cast metal ten cubits from brim to brim, circular in form, and its height was five cubits, and thirty cubits in circumference.
24Under its brim gourds went around encircling it ten to a cubit, completely surrounding the sea; the gourds were in two rows, cast with the rest.
Even the bible had an approximation for pi! =)
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Terms related to a circle
Segment of a circle – region on the circle bounded by a chord and an arc
Sector – region bounded by a central angle and an arc
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This sector is sooooooo delicious!
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MK
RK
P
PR
RKM
RPK
R K
P
M
Given circle P below, what name applies to each of the following?
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R K
P
M
Given circle P below, what name applies to each of the following?
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Indicate whether each statement is TRUE or FALSE.
Every diameter of a circle is a secant of the circle.
Every radius of a circle is a chord of the circle.
Every chord of a circle contains exactly two points of the circle.
If a radius bisects a chord of a circle, then it is to the chord.
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Indicate whether each statement is TRUE or FALSE.
The intersection of a line with a circle may be empty.
A line may intersect a circle in exactly one point.
The secant which is a perpendicular bisector of a chord of a circle contains the center of the circle.
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Insights on the Shortest Distance TSDB2P = The Shortest Distance Between 2Points
In geometry, TSDB2P is a straight line.In sickness, TSDB2P is relief.
Insights on the Shortest Distance TSDB2P = The Shortest Distance Between 2Points
In deep poverty, TSDB2P is realizing you have plenty to give. In a career, TSDB2P is integrity.In parenting, TSDB2P is allowing them to grow from their own mistakes.
Insights on the Shortest Distance TSDB2P = The Shortest Distance Between 2Points
In friendship, TSDB2P is trust.In learning, TSDB2P is a mind awaiting discovery. In personal growth, TSDB2P is learning your lesson the first time.
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Inscribed Angle Theorem
The measure of an inscribed angle is equal to one-half the degree measure of its intercepted arc.
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Central Angle Theorem
The central angle is twice the inscribed angle.
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Intersecting Chord Theorem
When two chords intersect each other inside a circle, the products of their segments are equal.
B
A C
D
E
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Angle Formed by Intersecting Chords
mABC =
____________________
B
A C
D
E
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Intersecting Secants Theorem
When two secant lines intersect each other outside a circle, the products of their segments are equal.
A
B
C
D
E
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Angle Formed by Intersecting Secants
mACE = ________________
A
B
C
D
E
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Intersecting Tangents Theorem
When two SEGMENTS are tangent to a circle, the segments are CONGRUENT.
AB
C
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Angle Formed by Intersecting Tangents
mBCA = ______________
AB
C
D
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Intersecting Tangent & Secant Theorem
When two secant lines intersect each other outside a circle, the products of their segments are equal.
A
B
C
D
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In the figure, M is the center of the circle.
Name a central angle.
Name a chord that is not a diameter
Name a major arc.
If mKMI = 170°, what is mKGI ?
What is mKHI ?
M
KG
I H
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Problem: Find the value of x.Given: Segment AB is tangent to circle C at B.
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Find the measure of each arc/angle: arc QSR Q R
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Name the arc/s intercepted by:x y z
xy
zR
Q
S
P
T
V
A
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Sometimes, secants intersect outside of circles. When this happens, the measure of the angle formed is equal to one-half the difference of the degree measures of the intercepted arcs.
Find the measure of angle 1.
Given: Arc AB = 60o Arc CD = 100o
Hmmm….Prove it first!
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Commonly Used Reasons
Radii of the same circle are congruent.
Base angles of an isosceles triangle are congruent.
The exterior angle is equal to the sum of the remote interior angles.
The degree measure of a minor arc is equal to the measure of the central angle which intercepts the arc.
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If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. (page 170)
P
x
R
Q
S
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PROOF:STATEMENTS REASONS
Given
Radii of the same circle are congruent
3. QRS is an isosceles Def. of an isosceles
4. PQR QRS Base angles of an isosceles are
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PROOF:STATEMENTS REASONS
5. mPQR = mQRS Def. of angles
6. mQRS = x Transitive Property
7. mPSR = 2x Central Angle Thm.
The minor arc ….
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PROOF:STATEMENTS REASONS
Transitive Property
Substitution
Division Property of Equality
Let’s learn & relearn...
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The line from the center of a circle to a chord ___________ the chord.bisects
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The measure
of the central
angle is
_________
the measure
of the
inscribed
angle
subtended by
the same arc.
twice
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The
inscribed
angles
subtended
by the
same arc
are
_________.congruent
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A tangent to a circle is ________________ to the radius at the point of tangency.
perpendicular
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Properties of Tangents
Tangent segments to a circle’s circumference from any external point are _______________.
The angle between the tangent and the chord is ____________________the intercepted arc.
half of the measure of
congruentThe angle between the tangent and the chord is _________________the inscribed angle on the opposite side of the chord.
equal to
Additional Exercises
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Find the value of x.
X
4
83
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2. Determine the length AC.
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3. Given a circle with the center O, and OF CD and DE = 20, OF = DF, find OF, EO, DF and DC.
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Find the length AC.
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Given that ED is tangent at C.
