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 = _______________