Chapter 10: Properties of Circles
Lesson 10.1: Use Properties of Tangents
Objectives Identify and label lines and segments related to the circle: chord, secant, tangent, and diameter.
Use properties of tangents to find measurements of chords and radii within a circle.
Example 1
Tell whether the line, ray, or segment is best
described as a radius, chord, diameter, secant,
or tangent of circle C.
AC = ________________ AB = _________________
DE = ________________ AE = _________________
Example 2
Use the diagram to find the given lengths.
Radius of circle A. _________ Diameter of circle B _________
Radius of circle C __________ Diameter of circle D _________
COPLANAR CIRCLES
Two circles who intersect in two points, one point, or no points.
COMMON TANGENT
A line, ray, or segment that is tangent to
two coplanar circles.
Example 3
Tell how many common tangents the circles have and use a ruler to draw them.
_____________________ ___________________ __________________
Tangent Theorem #1 Example 4
In a plane, a line is tangent to a circle if and only if the In the diagram, PT is the radius of circle P.
line is perpendicular to the radius of the circle at its Is ST tangent to circle P?
endpoint on the circle.
__________________________________
Tangent Theorem #2 Example 5
Tangent Segments from a common external RS is tangent to circle C at S and RT is
point are congruent. tangent to circle C at T. Find x.
x = ______________
Chapter 10: Properties of Circles
Lesson 10.2: Find Arc Measures
Objectives Formally define and explain the different angles and arcs in a circle: central angle along with its
corresponding major arc, minor arc, or semicircle.
Use properties of arcs to find their measure.
Determine the measure of central angles and their associated major and minor arcs.
NAMING ARCS
Minor Arcs _____________________________________
Major Arcs _____________________________________
MEASURING ARCS
The measure of the minor arc is equal to
_____________________________________________
The measure of the major arc is _________________________
____________________________________________________
The measure of a semicircle is _________________________
Example 1
Find the measure of each arc of circle P, where RT is the diameter.
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is
____________________________________________________ _____________________
Example 2
A recent survey asked teenagers if they would rather
meet a famous musician, athlete, actor, inventor, or
other person. The results are shown in the circle graph.
Find the indicated arc measures.
Example 3
_______________________ _______________________
Example 4
Tell which if any of the given arcs are congruent. Explain why or why not.
____________________ _____________________ _____________________
Example 5
A clock with hour and minutes hands is set to 1:00 p.m.
a) After 20 minutes, what will be the measure of the minor arc formed ____________________
by the hour and minute hands?
b) At what time before 2:00p.m., to the nearest minute, will the hour and minute __________________
hands form a diameter?
Chapter 10: Properties of Circles
Lesson 10.3: Apply Properties of Chords
Objectives Use properties of chords and arcs to find their measure.
Solve problems involving chords, radii and arcs within the same circle.
Chord Theorem #1 Example 1
In the same circle, or in congruent circles,
two minor arcs are congruent if and only if
their corresponding chords are congruent.
DCAB if and only if ___________________ ___________________________
BISECTING ARCS
A line, segment, or ray that cuts an arc into
two congruent arcs.
Chord Theorem #2 Chord Theorem #3
If one chord is the perpendicular bisector If a diameter of the circle is perpendicular to
of another chord, then the first chord is a chord, then the diameter bisects the chord
the diameter. and arc.
If QS is the perpendicular bisector of TR , If EG is a diameter and DFEG , then
___________________________________ ___________________________________
Example 2
Find the measure of the indicated arc in the diagram.
______________ ________________ _________________
AB ______________ ABD ________________
Chord Theorem #4 Example 3
In the same circle, or in congruent circles, two In the diagram of circle C, QR = ST = 16.
chords are congruent if and only if they are Find CU.
equidistant from the center.
_____________________ if and only if EF = EG CU = _____________________
Example 4
Determine whether AB is the diameter of the circle. Explain.
__________________________________________________________
__________________________________________________________
__________________________________________________________
Chapter 10: Properties of Circles
Lesson 10.4: Use Inscribed Angles and Polygons
Objectives Formally define and explain key terms such as inscribed angle and intercepted arc.
Use the Measure of an Inscribed Angle Theorem to solve problems involving inscribed angles
and their intercepted arcs.
Use the theorems about inscribed polygons to find the measure of the angles of an inscribed
polygon, along with the intercepted arcs.
Inscribed angle __________________________________________________
Intercepted arc __________________________________________________
ACTIVITY:
1) Use a ruler and draw a central angle. Label it <RPS.
2) Locate three points on P in the exterior of <RPS and
label them T, U, and V.
3) Draw <RTS, <RUS, and <RVS. These are inscribed angles.
Use a protractor and find the measure of each angle.
5) What do you notice about the measures of all the inscribed angles? ________________________
6) What do you notice about the measure of the inscribed angles compared to the corresponding central
angle? _________________________________________________
Measure of an Inscribed Angle Example 1
Theorem Find the indicated measure in circle P.
The measure of an inscribed angle is one
half the measure of its intercepted arc.
Central Angle Inscribed
Angle
Inscribed
Angle
Inscribed
Angle
Name <RPS <RTS <RUS <RVS
Measure
• P
Example 2
_____________ ______________
____________________________________________________________________________________
Inscribed Angle Theorem #2 Inscribed Triangle Theorem
If two inscribed angles intercept the same arc, If a right triangle is inscribed in a circle,
then…………. then …………
__________________________________ __________________________________
Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed inside of a circle
if and only if its opposite angles are supplementary.
m<E + m<G = _____________ and m<F + m<D = _____________
Example 3
Find the value of each variable.
x = _________ a = __________
y = _________ b = __________
Chapter 10: Properties of Circles
Lesson 10.5: Apply Other Angle Relationships in Circles
Objectives Use angles formed by tangents and chords to solve problems in geometry.
Use angles formed by lines that intersect inside, outside, and on a circle to solve problems.
Tangent-Chord Theorem
If a tangent and a chord intersect at a point on a circle,
then the measure of each angle formed is one half of the
measure of its intercepted arc.
Example 1
Find the indicated measure.
_________________ _________________ __________________
INTERSECTING LINES AND CIRCLES
Angles Inside the Circle Theorem
If two chords intersect inside a circle, then the measure of each
angle is one half the sum of the measures of the arcs intercepted
by the angle and its vertical angle.
Angles Outside the Circle Theorem
If a tangent and a secant, two tangents, or two secants intersect outside the circle, then the measure
of.the angle formed is
_______________________________________________________________________________
Example 2
Find the value of the variable.
.
x = __________________ a = __________________ x = __________________
Example 3
In the diagram, BA is tangent to circle E. Find
Example 4
Find the value of x.
= ________________
x = _________
Chapter 10: Properties of Circles
Lesson 10.6: Find Segment Lengths in Circles
Objectives Use various theorems to find the lengths of segments of chords within a circle.
Use various theorems to find the lengths of segments of tangents and secants within a circle.
Segments of Chords 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.
Segments of Secants 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.
Segments of Secants and Tangents Theorem
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.
Example 1 Example 2
Find the value of x. Find ML and JK.
x = _____________
ML = ___________
JK = ____________
x = ______________________
Example 3 Example 4
Find the value of x. Find the value of x.
x = ______________________ x = ______________________
Example 5
Find the value of x.
x = _______________ x = _______________