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USE PROPERTIES
OF TANGENTS
“THERE ARE NO SECRETS TO SUCCESS. IT IS
THE RESULT OF PREPARATION, HARD WORK,
AND LEARNING FROM FAILURE.” –COLIN
POWELL
CONCEPT 1: PARTS OF
A CIRCLE
Center-point equidistant from the edge of the circle.
Radius-segment from the center to the edge of the
circle.
Diameter-chord that passes through the center of the
circle.
radius diameter
center
CONCEPT 1: PARTS OF
A CIRCLE
Tangent-line that intersects the circle at only one point
called the point of tangency.
Secant-line that intersects the circle twice.
Chord-segment that connects two edges of the circle.
radius
Chord
center
Tangent
EXAMPLE 1
Match the notation with the term that best describes it.
1. 𝑨𝑩
2. 𝑩𝑯
3. 𝑫𝑩
4. 𝑬𝑮
5. Name the circle with notation.
A
B
C
D
E
F
G
H
CONCEPT 2: INTERSECTION
OF CIRCLES
Two circles can intersect in five main ways shown below.
Concentric circles are circles with the same center.
CONCEPT 3: COMMON
TANGENTS
A common tangent is a tangent line, ray or segment that is
tangent to multiple circles.
EXAMPLE 2
Draw the common tangents for circles that intersect only
once (tangent circles).
CONCEPT 4:
THEOREM 10.1
In a plane, a line is tangent to a circle iff the line is
perpendicular to a radius of the circle at its endpoint on the
circle (point of tangency).
radius
center
Tangent
EXAMPLE 3
Determine if 𝑨𝑩 is a tangent line. Explain how you know.
A
C
B
15
BC=17 8
EXAMPLE 4
Find the value of the radius. 𝑨𝑩 is a tangent.
A
C
B
48
36 r
r
CONCEPT 4:
THEOREM 10.2
Tangent segments form a common external point are
congruent.
Ex: If segment AC and segment BC are tangents, then
AC=BC.
B
A C
EXAMPLE 4
Find the value of x.
B
A C 𝑥2 − 4𝑥 + 2
𝑥2 − 4𝑥 − 2
“A DREAM DOESN’T BECOME REALITY THROUGH
MAGIC; IT TAKES SWEAT, DETERMINATION AND
HARD WORK.” –COLIN POWELL
FIND ARC
MEASURES
CONCEPT 5: ARCS AND
NOTATION
An arc of a circle is a portion of the circle. A minor arc
accounts for less than half the circle. A major arc
accounts for more than half the circle. A semicircle
accounts for exactly half of a circle.
Minor arc AB: 𝑨𝑩
Major arc from A through C to B: 𝑨𝑪𝑩
Semicircle from A through B to C: 𝑨𝑩𝑪
B A
C
CONCEPT 5: ARCS AND THEIR
MEASURE
The central angle is the angle formed by the endpoints of an
arc and the center of the circle. The measure of the arc is
equal to the measure of this angle.
m𝑨𝑩 =?
m𝑨𝑩𝑪 =?
m𝑨𝑪𝑩 =?
B A
C
80° 100°
D
Circle D
EXAMPLE 1
CONCEPT 6: ARC
ADDITION POSTULATE
The measure of an arc formed by two adjacent arcs is the
sum of the measures of the two arcs.
Ex: m𝑨𝑩 +m𝑩𝑪 =m𝑨𝑩𝑪
B A
C
80° 100°
D
Circle D
EXAMPLE 2
If the measure of arc AB is 79o and the measure of arc BC is
115o, what is the measure of arc ABC?
EXAMPLE 3
144°
36°
108°
72°
Missing Assignments
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
A
B
C
D
Find the measure of these arcs:
1. m𝐴𝐵𝐶
2. m𝐴𝐵𝐷
3. m𝐷𝐵𝐶
4. m𝐷𝐴𝐵
5. m𝐶𝐷𝐴
CONCEPT 7: CONGRUENT
CIRCLES AND ARCS
Circles are congruent if they have the same radius.
(consequently if they have the same diameter)
Arcs are congruent if they have the same measure and are of
the same circle or congruent circles.
EXAMPLE 4
APPLY
PROPERTIES
OF CHORDS
“INDIVIDUAL COMMITMENT TO A GROUP EFFORT –
THAT IS WHAT MAKES A TEAM WORK, A COMPANY
WORK, A SOCIETY WORK, A CIVILIZATION WORK.” –
VINCE LOMBARDI
CONCEPT 8: THEOREM 10.3
In the same circle or in congruent circles, two minor arcs are
congruent iff their corresponding chords are congruent.
EXAMPLE 1
Find the measure of the arc given information provided
below.
RS=ST and m𝑹𝑻 =70°
Find m𝑹𝑺 .
EXAMPLE 2
Find the measure of the arc given information provided
below.
RS=ST and m𝑺𝑻 =70°
Find m𝑹𝑺 .
CONCEPT 9: THEOREM 10.4
If one chord is a perpendicular bisector of another chord,
then the first chord is a diameter.
diameter
EXAMPLE 3
CONCEPT 9: THEOREM 10.5
If a diameter of a circle is perpendicular to a chord, then the
diameter bisects the chord and its arc.
EXAMPLE 4
P
P
P
CONCEPT 10: THEOREM
10.6
In the same circle, or in congruent circles, two chords are
congruent iff they are equidistant from the center.
EXAMPLE 5
Find QR, QU, and the radius of circle C if ST=32, UC =12, and
CV=12.
USE INSCRIBED
ANGLES AND
POLYGONS
“CHOOSE A JOB YOU LOVE, AND YOU WILL
NEVER HAVE TO WORK A DAY IN YOUR
LIFE.” –CONFUCIUS
CONCEPT 11: MEASURE OF
INSCRIBED ANGLE
An inscribed angle is an angle whose vertex is on the edge of
the circle and its sides are chords.
Intercepted arc is the arc formed by an inscribed angle.
The measure of an inscribed angle is one half the measure of
the intercepted arc.
Inscribed angle
A
B
Intercepted arc
EXAMPLE 1
CONCEPT 12:
THEOREM 10.8
If two inscribed angles of a circle intercept the same arc, then
the angles are congruent.
CONCEPT 13: INSCRIBED
POLYGONS
An inscribed polygon has all of vertices on the edge of
another polygon/shape.
A circumscribed circle is a circle that contains a
polygon/shape on the inside.
CONCEPT 14:
THEOREM 10.9
Part 1: If a right triangle is inscribed in a circle, then the
hypotenuse will be a diameter of the circle.
Part 2: If one side of an inscribed triangle is the diameter of
the circle, then the triangle is a right triangle and the angle
opposite the diameter is the right angle.
diameter
EXAMPLE 3
Find the value of each variable.
CONCEPT 14:
THEOREM 10.10
A quadrilateral can be inscribed in a circle iff its opposite
angles are supplementary.
A
B
C
D
EXAMPLE 4
Find the value of each variable.
EXAMPLE 5
Which shapes can ALWAYS be inscribed in a circle? Explain.
1) Rectangle
2) Right Triangle
3) Isosceles Triangle
4) Scalene Triangle
5) Parallelogram
6) Kite
7) Trapezoid
8) Isosceles Trapezoid
APPLY OTHER
ANGLE
RELATIONSHIPS
IN CIRCLES
“OPPORTUNITY IS MISSED BY MOST PEOPLE BECAUSE IT
IS DRESSED IN OVERALLS AND LOOKS LIKE WORK.”
–THOMAS A. EDISON
LINES INTERSECTING
CONCEPT 15: THEOREM
10.11
If a tangent and a chord intersect at a point on a circle, then
the measure of each angle formed is one half the measure of
its intercepted arc.
EXAMPLE 1
CONCEPT 16: ANGLES INSIDE
OF THE CIRCLE
If two chords (or two secants) intersect inside a circle, then
the measure of each angle is one half the sum of the measure
of the arcs intercepted by the angle and its vertical angle.
EXAMPLE 2
Find the value of z.
CONCEPT 17: ANGLES OUTSIDE
THE CIRCLE
If a tangent and a secant, two tangents, or two secants
intersect outside a circle, then the measure of the angle
formed is one half the difference of the measures of the
intercepted arcs.
EXAMPLE 3
Find the value of the variable.
EXAMPLE 4
FIND SEGMENT
LENGTHS IN
CIRCLES
“THE BEST PREPARATION FOR GOOD
WORK TOMORROW IS TO DO GOOD WORK
TODAY.” –ELBERT HUBBARD
CONCEPT 18: SEGMENTS OF
CHORDS
Segments of chords are smaller segments that make up a
chord.
Theorem: If two chords intersect in the interior of a circle,
then the product of the lengths of the segments of one chord
is equal to the product of the lengths of the segments of the
other chord.
Ex: bd=ac
EXAMPLE 1
Find the value of x.
CONCEPT 19: SEGMENTS OF
SECANTS
Secant segment is a segment that contains a chord of a
circle and has only one endpoint outside the circle.
The external segment of a secant segment is the portion of
the secant segment outside the circle.
Theorem: If two secant segments share the same endpoint
outside a circle, then the product of the lengths of one secant
segment and its external segment equals the product of the
lengths of the other secant segment and its external
segment.
Ex: b(b+a)=c(c+d)
EXAMPLE 2
Find the value of x.
EXAMPLE 3
CONCEPT 20: SEGMENTS OF
TANGENTS AND SECANTS
If a secant segment and a tangent segment share an
endpoint outside a circle, then the product of the lengths of
the secant segment and its external segment equals the
square of the length of the tangent segment.
Ex: y(y+x)=z(z)
y(y+x)=𝒛𝟐
EXAMPLE 4
Find the value of x.
EXAMPLE 5
WRITE AND
GRAPH
EQUATIONS OF
CIRCLES
“THE ONLY PLACE SUCCESS COMES
BEFORE WORK IS IN THE DICTIONARY.” –
VINCE LOMBARDI
CONCEPT 21: STANDARD
EQUATION OF A CIRCLE
The standard equation of a circle with center (h, k) and radius
r is: (𝒙 − 𝒉)𝟐+(𝒚 − 𝒌)𝟐= 𝒓𝟐.
EXAMPLE 1
Write the standard equation of the circle.
1. Center (-5, 2) and radius 3
2. Center (2, -4) and radius 4
3. Center (0, 0) and radius 1
CONCEPT 22: GRAPH
EQUATIONS OF A CIRCLE
To graph an equation of a circle:
1. Figure out the center and radius of the circle
2. Plot the center of the circle in the graph
3. Measure the radius right, left, up, and down.
1. Create 4 new points with those measures
4. Connect the four dots with arcs.
EXAMPLE 2
Graph these equations of a circle.
CONCEPT 23: WRITE
EQUATION FROM A GRAPH
To write an equation from a graph:
1. Identify the center and radius of the circle.
2. Plug the center and radius into the standard equation of a
circle.
3. Square r.
EXAMPLE 3
Write the equation for each circle.