Lesson 10-1 Circles and Circumferences

Lesson 10-2 Angles and Arcs

Lesson 10-3 Arcs and Chords

Lesson 10-4 Inscribed Angles

Lesson 10-5 Tangents

Lesson 10-6 Secants, Tangents, and Angle Measures

Lesson 10-7 Special Segments in a Circle

Lesson 10-8 Equations of Circles

Example 1 Identify Parts of a Circle

Example 2 Find Radius and Diameter

Example 3 Find Measures in Intersecting Circles

Example 4Find Circumference, Diameter, and Radius

Example 5 Use Other Figures to Find Circumference

Name the circle.

Answer: The circle has its center at E, so it is named circle E, or .

Answer: Four radii are shown: .

Name the radius of the circle.

Answer: Four chords are shown: .

Name a chord of the circle.

Name a diameter of the circle.

Answer: are the only chords that go through the center. So, are diameters.

Answer:

Answer:

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.

Answer:

Answer:

Answer: 9

Formula for radius

Substitute and simplify.

If ST 18, find RS.Circle R has diameters and .

Answer: 48

Formula for diameter

Substitute and simplify.

If RM 24, find QM.Circle R has diameters .

Answer: So, RP = 2.

Since all radii are congruent, RN = RP.

If RN 2, find RP.Circle R has diameters .

Answer: 58

Answer: 12.5

a. If BG = 25, find MG.

b. If DM = 29, find DN.

Circle M has diameters

c. If MF = 8.5, find MG.

Answer: 8.5

Find EZ.

The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively.

Since the diameter of FZ = 5.

Since the diameter of , EF = 22.

Segment Addition Postulate

Substitution

is part of .

Simplify.

Answer: 27 mm

Find XF.

The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively.

Since the diameter of , EF = 22.

Answer: 11 mm

is part of . Since is a radius of

The diameters of , and are 5 inches, 9 inches, and 18 inches respectively.a. Find AC.

b. Find EB.

Answer: 6.5 in.

Answer: 13.5 in.

Find C if r = 13 inches.

Circumference formula

Substitution

Answer:

Find C if d = 6 millimeters.

Circumference formula

Substitution

Answer:

Find d and r to the nearest hundredth if C = 65.4 feet.

Circumference formula

Substitution

Use a calculator.

Divide each side by .

Radius formula

Use a calculator.

Answer:

a. Find C if r = 22 centimeters.

b. Find C if d = 3 feet.

c. Find d and r to the nearest hundredth if C = 16.8 meters.

Answer:

Answer:

Answer:

Read the Test ItemYou are given a figure that involves a right triangle and a circle. You are asked to find the exact circumference of the circle.

MULTIPLE- CHOICE TEST ITEM Find the exact circumference of . A B C D

Solve the Test ItemThe radius of the circle is the same length as either leg of the triangle. The legs of the triangle have equal length. Call the length x.

Pythagorean Theorem

Substitution

Divide each side by 2.

Simplify.

Take the square root of each side.

So the radius of the circle is 3.

Circumference formula

Substitution

Because we want the exact circumference, the answer is B.

Answer: B

Answer: C

Find the exact circumference of . A B C D

Example 1 Measures of Central Angles

Example 2 Measures of Arcs

Example 3 Circle Graphs

Example 4 Arc Length

ALGEBRA Refer to .Find .

SubstitutionSimplify.Add 2 to each side.Divide each side by 26.

Use the value of x to findGivenSubstitution

Answer: 52

The sum of the measures of

ALGEBRA Refer to .Find .

Linear pairs are supplementary.

Substitution

Simplify.

Subtract 140 from each side.

form a linear pair.

Answer: 40

Answer: 65

Answer: 40

ALGEBRA Refer to .

a. Find m

b. Find m

Find .

In bisects and

is a minor arc, sois a semicircle.

is a right angle.

Arc Addition PostulateSubstitutionSubtract 90 from each side.

Answer: 90

Find .

In bisects and

since bisects .

is a semicircle.

Arc Addition Postulate

Subtract 46 from each side.

Answer: 67

Find .

In bisects and

Vertical angles are congruent.Substitution.

Substitution.Subtract 46 from each side.

Subtract 44 from each side.Substitution.

Answer: 316

Answer: 54

Answer: 72

In and are diameters, and bisects Find each measure.

a.

b.

c. Answer: 234

BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001.Find the measurement of the central angle representing each category. List them from least to greatest.

The sum of the percents is 100% and represents the whole. Use the percents to determine what part of the whole circle each central angle contains.

Answer:

BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001.Is the arc for the wedge named Youth congruent to the arc for the combined wedges named Other and Comfort?

Answer: no

The arc for the wedge named Youth represents 26% or of the circle. The combined wedges named Other and Comfort represent

. Since º, the arcs are not congruent.

SPEED LIMITS This graph shows the percent of U.S. states that have each speed limit on their interstate highways.

Answer: no

b. Is the arc for the wedge for 65 mph congruent to the combined arcs for the wedges for 55 mph and 70 mph?

a. Find the measurement of the central angles representing each category. List them from least to greatest.

Answer:

In and . Find the length of .

In and . Write a proportion to compare each part to its whole.

Now solve the proportion for .

Simplify.

Answer: The length of is units or about 3.14 units.

degree measure of arcdegree measure of

whole circle

arc lengthcircumference

Multiply each side by 9 .

In and . Find the length of .

Answer: units or about 49.48 units

Example 1 Prove Theorems

Example 2 Inscribed Polygons

Example 3 Radius Perpendicular to a Chord

Example 4 Chords Equidistant from Center

PROOF Write a proof.

Prove:

Given:

is a semicircle.

Proof:Statements Reasons1. 1. Given

is a semicircle.

5. Def. of arc measure5.

2. Def. of semicircle2. 3. In a circle, 2 chords

are , corr. minor arcs are .

3.

4. Def. of arcs4.

Answer:Statements Reasons6. 6. Arc Addition Postulate7. 7. Substitution8. 8. Subtraction Property

and simplify9. 9. Division Property10. 10. Def. of arc measure11. 11. Substitution

PROOF Write a proof.

Prove:

Given:

Proof:Statements Reasons1.2.

3.4.

1. Given2. In a circle, 2 minor

arcs are , chords are .

3. Transitive Property4. In a circle, 2 chords

are , minor arcs are .

TESSELLATIONS The rotations of a tessellation can create twelve congruent central angles. Determine whether .

Answer: Since the measures of are equal, .

Because all of the twelve central angles are congruent, the measure of each angle is

Let the center of the circle be A. The measure of Then .

The measure of Then .

ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether .

Answer: no

Circle W has a radius of 10 centimeters. Radius is perpendicular to chord which is 16 centimeters long.If find

Since radius is perpendicular to chord

Arc addition postulate

Substitution

Substitution

Subtract 53 from each side.

Answer: 127

Circle W has a radius of 10 centimeters. Radius is perpendicular to chord which is 16 centimeters long.Find JL.

A radius perpendicular to a chord bisects it.

Definition of segment bisector

Draw radius

Use the Pythagorean Theorem to find WJ.

Pythagorean Theorem

Simplify.

Subtract 64 from each side.

Take the square root of each side.

Segment addition

Subtract 6 from each side.

Answer: 4

Answer: 145

Answer: 10

Circle O has a radius of 25 units. Radius is perpendicular to chord which is 40 units long.

a. If

b. Find CH.

Chords and are equidistant from the center. If the radius of is 15 and EF = 24, find PR and RH.

are equidistant from P, so .

Draw to form a right triangle. Use the Pythagorean Theorem.

Pythagorean Theorem

Simplify.

Subtract 144 from each side.

Take the square root of each side.

Answer:

Answer:

Chords and are equidistant from the center of If TX is 39 and XY is 15, find WZ and UV.

Example 1 Measures of Inscribed Angles

Example 2 Proofs with Inscribed Angles

Example 3 Inscribed Arcs and Probability

Example 4 Angles of an Inscribed Triangle

Example 5 Angles of an Inscribed Quadrilateral

In and Find the measures of the numbered angles.

Arc Addition Theorem

Simplify.

Subtract 168 from each side.Divide each side by 2.

First determine

So, m

Answer:

In and Find the measures of the numbered angles.

Answer:

Given:

Prove:

Proof:Statements Reasons

1. Given1.2. 2. If 2 chords are , corr.

minor arcs are .3. 3. Definition of

intercepted arc4. 4. Inscribed angles of

arcs are .5. 5. Right angles are

congruent6. 6. AAS

Prove:

Given:

1. Given2. Inscribed angles of

arcs are .3. Vertical angles are

congruent.4. Radii of a circle are

congruent.5. ASA

Proof:Statements Reasons1.2.

3.

4.

5.

PROBABILITY Points M and N are on a circle so that . Suppose point L is randomly located on the same circle so that it does not coincide with M or N. What is the probability that

Since the angle measure is twice the arc measure, inscribed must intercept , so L must lie on minor arc MN. Draw a figure and label any information you know.

The probability that is the same as the probability of L being contained in .

Answer: The probability that L is located on is

PROBABILITY Points A and X are on a circle so that Suppose point B is randomly located on the same circle so that it does not coincide with A or X. What is the probability that

Answer:

ALGEBRA Triangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and

are right triangles. since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so .

Angle Sum Theorem

Simplify.

Subtract 105 from each side.

Divide each side by 3.

Use the value of x to find the measures of

Given Given

Answer:

Answer:

ALGEBRA Triangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and

Quadrilateral QRST is inscribed in If and find and

Draw a sketch of this situation.

To find we need to know

To find first find

Inscribed Angle Theorem

Sum of angles in circle = 360

Subtract 174 from each side.

Inscribed Angle TheoremSubstitutionDivide each side by 2.

To find we need to know but first we must find

Inscribed Angle Theorem

Sum of angles in circle = 360

Subtract 204 from each side.

Inscribed Angle Theorem

Divide each side by 2.

Answer:

Answer:

Quadrilateral BCDE is inscribed in If and find and

Example 1 Find Lengths

Example 2 Identify Tangents

Example 3 Solve a Problem Involving Tangents

Example 4 Triangles Circumscribed About a Circle

ALGEBRA is tangent to at point R. Find y.

Because the radius is perpendicular to the tangent at the point of tangency, . This makes a right angle and a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

Pythagorean Theorem

Simplify.

Subtract 256 from each side.Take the square root of each side.

Because y is the length of the diameter, ignore the negative result.

Answer: Thus, y is twice .

Answer: 15

is a tangent to at point D. Find a.

First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem.

Determine whether is tangent to

Pythagorean Theorem

Simplify.

Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle.

Answer: So, is not tangent to .

First determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem.

Determine whether is tangent to

Pythagorean Theorem

Simplify.

Answer: Thus, making a tangent to

Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle.

Answer: yes

a. Determine whether is tangent to

Answer: no

b. Determine whether is tangent to

ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to

Definition of congruent segments

Substitution.

Use the value of y to find x.Definition of congruent segments

Substitution

Simplify.

Subtract 14 from each side.

Answer: 1

ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent.

Answer: –6

Triangle HJK is circumscribed about Find the perimeter of HJK if

Use Theorem 10.10 to determine the equal measures.

We are given that

Answer: The perimeter of HJK is 158 units.

Definition of perimeter

Substitution

Triangle NOT is circumscribed about Find the perimeter of NOT if

Answer: 172 units

Example 1 Secant-Secant Angle

Example 2 Secant-Tangent Angle

Example 3 Secant-Secant Angle

Example 4 Tangent-Tangent Angle

Example 5 Secant-Tangent Angle

Find if and

Method 1

Method 2

Answer: 98

Answer: 138

Find if and

Find if and

Answer: 55

Answer: 58

Find if and

Find x.

Theorem 10.14

Multiply each side by 2.Add x to each side.Subtract 124 from each side.

Answer: 17

Find x.

Answer: 111

JEWELRY 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.

Let x represent the measure of the arc at the bottom of the pendant. Then the arc at the top of the circle will be 360 – x. The measure of the angle marked 40° is equal to one-half the difference of the measure of the

two intercepted arcs.

Multiply each side by 2 and simplify.

Add 360 to each side.

Divide each side by 2.

Answer: 220

Answer: 75

PARKS Two sides of a fence to be built around a circular garden in a park are shown. Use the figure to determine the measure of

Find x.

Multiply each side by 2.

Add 40 to each side.

Divide each side by 6.

Answer: 25

Find x.

Answer: 9

Example 1 Intersection of Two Chords

Example 2 Solve Problems

Example 3 Intersection of Two Secants

Example 4 Intersection of a Secant and a Tangent

Find x.

Theorem 10.15

Multiply.

Divide each side by 8.

Answer: 13.5

Find x.

Answer: 12.5

BIOLOGY 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 if it is located 0.25 mm from the bottom of the field of view. Round to the nearest hundredth.

Draw a model using a circle. Let x represent the unknown measure of the equal lengths of the chord which is the length of the organism. Use the products of the lengths of the intersecting chords to find the length of the organism. Note that

Segment products

Substitution

Simplify.

Answer: 0.66 mm

Take the square root of each side.

ARCHITECTURE Phil is installing a new window in an addition for a client’s home. The window is a rectangle with an arched top called an eyebrow. The diagram below shows the dimensions of the window. What is the radius of the circle containing the arc if the eyebrow portion of the window is not a semicircle?

Answer: 10 ft

Find x if EF 10, EH 8, and FG 24.

Secant Segment Products

Substitution

Distributive Property

Subtract 64 from each side.

Divide each side by 8.

Answer: 34.5

Answer: 26

Find x if and

Answer: 8

Find x. Assume that segments that appear to be tangent are tangent.

Disregard the negative solution.

Find x. Assume that segments that appear to be tangent are tangent.

Answer: 30

Example 1 Equation of a Circle

Example 2 Use Characteristics of Circles

Example 3 Graph a Circle

Example 4 A Circle Through Three Points

Equation of a circle

Simplify.

Answer:

Write an equation for a circle with the center at(3, –3), d 12.

Equation of a circle

Simplify.

Answer:

Write an equation for a circle with the center at(–12, –1), r 8.

Answer:

Write an equation for each circle.

a. center at (0, –5), d 18

b. center at (7, 0), r 20

Answer:

Sketch a drawing of the two tangent lines.

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.

Since d 10, r 5. The line x –1 is perpendicular to a radius. Since x –1 is a vertical line, the radius lies on a horizontal line. Count 5 units to the right from x –1. Find the value of h.

The center is at (4, 2), and the radius is 5.

Answer: An equation for the circle is .

Likewise, the radius perpendicular to the line y –3 lies on a vertical line. The value of k is 5 units up from –3.

Answer:

A circle with a diameter of 8 has its center in the second quadrant. The lines y –1 and x 1 are tangent to the circle. Write an equation of the circle.

GraphCompare each expression in the equation to the standard form.

The center is at (2, –3), and the radius is 2.Graph the center. Use a compass set at a width of 2 grid squares to draw the circle.

Answer:

Graph

Write the expression in standard form.

The center is at (3, 0), and the radius is 4.

Draw a circle with radius 4, centered at (3, 0).

Answer:

Answer:

a. Graph

Answer:

b. Graph

ELECTRICITY Strategically located substations are extremely important in the transmission and distribution of a power company’s electric supply. Suppose three substations are modeled by the points D(3, 6), E(–1, 0), and F(3, –4). Determine the location of a town equidistant from all three substations, and write an equation for the circle.

Explore You are given three points that lie on a circle.Plan Graph DEF. Construct the perpendicular

bisectors of two sides to locate the center, which is the location of the tower. Find the length of a radius. Use the center and radius to write an equation.

Solve Graph DEF and construct the perpendicular bisectors of two sides. The center appears to be at (4, 1). This is the location of the tower. Find r by using the Distance Formula with the center and any of the three points.

Write an equation.

Examine You can verify the location of the center by finding the equations of the two bisectors and solving a system of equations. You can verify the radius by finding the distance between the center and another of the three points on the circle.

Answer:

AMUSEMENT PARKS The designer of an amusement park wants to place a food court equidistant from the roller coaster located at (4, 1), the Ferris wheel located at (0, 1), and the boat ride located at (4, –3). Determine the location for the food court and write an equation for the circle.

Answer: