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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 Contents

Example 1 Identify Parts of a Circle Example 2 Find Radius and Diameter Example 3 Find Measures in Intersecting Circles Example 4 Find Circumference, Diameter, and Radius Example 5 Use Other Figures to Find Circumference Lesson 1 Contents

Answer: The circle has its center at E, so it is named circle E, or . Name the circle. Answer: The circle has its center at E, so it is named circle E, or . Example 1-1a

Name the radius of the circle. Answer: Four radii are shown: . Example 1-1b

Name a chord of the circle. Answer: Four chords are shown: . Example 1-1c

Name a diameter of the circle. Answer: are the only chords that go through the center. So, are diameters. Example 1-1d

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: Answer: Example 1-1e

Circle R has diameters and . If ST 18, find RS. Circle R has diameters and . Formula for radius Substitute and simplify. Answer: 9 Example 1-2a

Substitute and simplify. If RM 24, find QM. Circle R has diameters . Formula for diameter Substitute and simplify. Answer: 48 Example 1-2b

Since all radii are congruent, RN = RP. If RN 2, find RP. Circle R has diameters . Since all radii are congruent, RN = RP. Answer: So, RP = 2. Example 1-2c

Circle M has diameters a. If BG = 25, find MG. Answer: 12.5 b. If DM = 29, find DN. Circle M has diameters c. If MF = 8.5, find MG. Answer: 12.5 Answer: 58 Answer: 8.5 Example 1-2d

The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively. Find EZ. Example 1-3a

Since the diameter of , EF = 22. Since the diameter of FZ = 5. is part of . Segment Addition Postulate Substitution Simplify. Answer: 27 mm Example 1-3b

Find XF. The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively. Example 1-3c

Since the diameter of , EF = 22. is part of . Since is a radius of Answer: 11 mm Example 1-3d

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. Example 1-3e

Circumference formula Find C if r = 13 inches. Circumference formula Substitution Answer: Example 1-4a

Find C if d = 6 millimeters. Circumference formula Substitution Answer: Example 1-4b

Find d and r to the nearest hundredth if C = 65.4 feet. Circumference formula Substitution Divide each side by . Use a calculator. Example 1-4c

Radius formula Use a calculator. Answer: Example 1-4d

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: Example 1-4e

MULTIPLE- CHOICE TEST ITEM Find the exact circumference of . A B C D Read the Test Item You are given a figure that involves a right triangle and a circle. You are asked to find the exact circumference of the circle. Example 1-5a

Take the square root of each side. Solve the Test Item The 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 Simplify. Divide each side by 2. Take the square root of each side. Example 1-5b

So the radius of the circle is 3. Circumference formula Substitution Because we want the exact circumference, the answer is B. Answer: B Example 1-5c

Find the exact circumference of . A B C D Answer: C Example 1-5d

End of Lesson 1

Example 1 Measures of Central Angles Example 2 Measures of Arcs Example 3 Circle Graphs Example 4 Arc Length Lesson 2 Contents

ALGEBRA Refer to . Find . Example 2-1a

The sum of the measures of Substitution Simplify. Add 2 to each side. Divide each side by 26. Use the value of x to find Given Substitution Answer: 52 Example 2-1b

ALGEBRA Refer to . Find . Example 2-1c

Linear pairs are supplementary. form a linear pair. Linear pairs are supplementary. Substitution Simplify. Subtract 140 from each side. Answer: 40 Example 2-1d

ALGEBRA Refer to . a. Find m b. Find m Answer: 65 Answer: 40 Example 2-1e

In bisects and Find . Example 2-2a

Arc Addition Postulate is a minor arc, so is a semicircle. is a right angle. Arc Addition Postulate Substitution Subtract 90 from each side. Answer: 90 Example 2-2b

In bisects and Find . Example 2-2c

Arc Addition Postulate since bisects . is a semicircle. Arc Addition Postulate Subtract 46 from each side. Answer: 67 Example 2-2d

In bisects and Find . Example 2-2e

Vertical angles are congruent. Substitution. Substitution. Subtract 46 from each side. Substitution. Subtract 44 from each side. Answer: 316 Example 2-2f

In and are diameters, and bisects Find each measure. a. Answer: 54 Answer: 72 Answer: 234 Example 2-2g

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. Example 2-3a

The sum of the percents is 100% and represents the whole 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: Example 2-3b

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? Example 2-3c

The arc for the wedge named Youth represents 26% or of the circle 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. Answer: no Example 2-3d

SPEED LIMITS This graph shows the percent of U. S SPEED LIMITS This graph shows the percent of U.S. states that have each speed limit on their interstate highways. Example 2-3e

b. Is the arc for the wedge for 65 mph congruent to the 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: Answer: no Example 2-3f

In and . Find the length of . In and . Write a proportion to compare each part to its whole. Example 2-4a

degree measure of whole circle arc length circumference degree measure of arc degree measure of whole circle arc length circumference Now solve the proportion for . Multiply each side by 9 . Simplify. Answer: The length of is units or about 3.14 units. Example 2-4b

In and . Find the length of . Answer: units or about 49.48 units Example 2-4c

End of Lesson 2

Example 1 Prove Theorems Example 2 Inscribed Polygons Example 3 Radius Perpendicular to a Chord Example 4 Chords Equidistant from Center Lesson 3 Contents

PROOF Write a proof. Prove: Given: is a semicircle. Example 3-1a

3. In a circle, 2 chords are , corr. minor arcs are . 3. Proof: Statements Reasons 1. 1. Given is a semicircle. 2. Def. of semicircle 2. 3. In a circle, 2 chords are , corr. minor arcs are . 3. 4. Def. of arcs 4. 5. Def. of arc measure 5. Example 3-1b

6. Arc Addition Postulate Answer: Statements Reasons 6. 6. Arc Addition Postulate 7. 7. Substitution 8. 8. Subtraction Property and simplify 9. 9. Division Property 10. 10. Def. of arc measure 11. 11. Substitution Example 3-1c

PROOF Write a proof. Given: Prove: Example 3-1d

2. In a circle, 2 minor arcs are , chords are . Proof: Statements Reasons 1. 2. 3. 4. 1. Given 2. In a circle, 2 minor arcs are , chords are . 3. Transitive Property 4. In a circle, 2 chords are , minor arcs are . Example 3-1e

TESSELLATIONS The rotations of a tessellation can create twelve congruent central angles. Determine whether . Example 3-2a

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 . Answer: Since the measures of are equal, . Example 3-2b

ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether . Answer: no Example 3-2c

Circle W has a radius of 10 centimeters Circle W has a radius of 10 centimeters. Radius is perpendicular to chord which is 16 centimeters long. If find Example 3-3a

Since radius is perpendicular to chord Arc addition postulate Substitution Substitution Subtract 53 from each side. Answer: 127 Example 3-3b

Circle W has a radius of 10 centimeters Circle W has a radius of 10 centimeters. Radius is perpendicular to chord which is 16 centimeters long. Find JL. Example 3-3c

A radius perpendicular to a chord bisects it. Draw radius  A radius perpendicular to a chord bisects it. Definition of segment bisector Example 3-3d

Use the Pythagorean Theorem to find WJ. Simplify. Subtract 64 from each side. Take the square root of each side. Example 3-3e

Subtract 6 from each side. Segment addition Subtract 6 from each side. Answer: 4 Example 3-3f

Circle O has a radius of 25 units Circle O has a radius of 25 units. Radius is perpendicular to chord which is 40 units long. a. If b. Find CH. Answer: 145 Answer: 10 Example 3-3g

Chords and are equidistant from the center Chords and are equidistant from the center. If the radius of is 15 and EF = 24, find PR and RH. Example 3-4a

are equidistant from P, so . Example 3-4b

Draw to form a right triangle. Use the Pythagorean Theorem. Simplify. Subtract 144 from each side. Take the square root of each side. Answer: Example 3-4c

Chords and are equidistant from the center of If TX is 39 and XY is 15, find WZ and UV. Answer: Example 3-4d

End of Lesson 3

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 Lesson 4 Contents

In and Find the measures of the numbered angles. Example 4-1a

Subtract 168 from each side. First determine Arc Addition Theorem Simplify. Subtract 168 from each side. Divide each side by 2. Example 4-1b

So, m Example 4-1c

Answer: Example 4-1d

In and Find the measures of the numbered angles. Answer: Example 4-1e

Given: Prove: Example 4-2a

2. If 2 chords are , corr. minor arcs are . Proof: Statements Reasons 1. Given 1. 2. 2. If 2 chords are , corr. minor arcs are . 3. 3. Definition of intercepted arc 4. 4. Inscribed angles of arcs are . 5. 5. Right angles are congruent 6. 6. AAS Example 4-2b

Given: Prove: Example 4-2c

2. Inscribed angles of arcs are . 3. Vertical angles are congruent. 1. Given 2. Inscribed angles of arcs are . 3. Vertical angles are congruent. 4. Radii of a circle are congruent. 5. ASA Proof: Statements Reasons 1. 2. 3. 4. 5. Example 4-2d

PROBABILITY Points M and N are on a circle so that 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. Example 4-3a

Answer: The probability that L is located on is The probability that is the same as the probability of L being contained in . Answer: The probability that L is located on is Example 4-3b

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: Example 4-3c

ALGEBRA Triangles TVU and TSU are inscribed in with ALGEBRA Triangles TVU and TSU are inscribed in with Find the measure of each numbered angle if and Example 4-4a

Subtract 105 from each side. 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. Example 4-4b

Use the value of x to find the measures of Given Given Answer: Example 4-4c

ALGEBRA Triangles MNO and MPO are inscribed in with ALGEBRA Triangles MNO and MPO are inscribed in with Find the measure of each numbered angle if and Answer: Example 4-4d

Quadrilateral QRST is inscribed in If and find and Draw a sketch of this situation. Example 4-5a

Inscribed Angle Theorem To find we need to know To find first find Inscribed Angle Theorem Sum of angles in circle = 360 Subtract 174 from each side. Example 4-5b

Inscribed Angle Theorem Substitution Divide each side by 2. To find we need to know but first we must find Inscribed Angle Theorem Example 4-5c

Sum of angles in circle = 360 Subtract 204 from each side. Inscribed Angle Theorem Divide each side by 2. Answer: Example 4-5d

Quadrilateral BCDE is inscribed in If and find and Answer: Example 4-5e

End of Lesson 4

Example 2 Identify Tangents Example 1 Find Lengths Example 2 Identify Tangents Example 3 Solve a Problem Involving Tangents Example 4 Triangles Circumscribed About a Circle Lesson 5 Contents

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. Example 5-1a

Subtract 256 from each side. 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 . Example 5-1b

is a tangent to at point D. Find a. Answer: 15 Example 5-1c

Determine whether is tangent to First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem. Example 5-2a

Answer: So, is not 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 . Example 5-2b

Determine whether is tangent to First determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem. Example 5-2c

Answer: Thus, making a tangent to Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle. Answer: Thus, making a tangent to Example 5-2d

a. Determine whether is tangent to Answer: yes Example 5-2e

b. Determine whether is tangent to Answer: no Example 5-2f

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 Example 5-3a

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 Example 5-3b

ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Answer: –6 Example 5-3d

Triangle HJK is circumscribed about Find the perimeter of HJK if Example 5-4a

Use Theorem 10.10 to determine the equal measures. We are given that Definition of perimeter Substitution Answer: The perimeter of HJK is 158 units. Example 5-4b

Triangle NOT is circumscribed about Find the perimeter of NOT if Answer: 172 units Example 5-4c

End of Lesson 5

Example 1 Secant-Secant Angle Example 2 Secant-Tangent Angle Example 4 Tangent-Tangent Angle Example 5 Secant-Tangent Angle Lesson 6 Contents

Find if and Method 1 Example 6-1a

Method 2 Answer: 98 Example 6-1b

Find if and Answer: 138 Example 6-1d

Find if and Answer: 55 Example 6-2a

Find if and Answer: 58 Example 6-2c

Subtract 124 from each side. Find x. Theorem 10.14 Multiply each side by 2. Add x to each side. Subtract 124 from each side. Answer: 17 Example 6-3a

Find x. Answer: 111 Example 6-3c

JEWELRY A jeweler wants to craft a pendant with the shape shown 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. Example 6-4a

Multiply each side by 2 and simplify. Add 360 to each side. Divide each side by 2. Answer: 220 Example 6-4b

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 Answer: 75 Example 6-4c

Find x. Multiply each side by 2. Add 40 to each side. Divide each side by 6. Answer: 25 Example 6-5a

Find x. Answer: 9 Example 6-5c

End of Lesson 6

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 Lesson 7 Contents

Find x. Theorem 10.15 Multiply. Divide each side by 8. Answer: 13.5 Example 7-1a

Find x. Answer: 12.5 Example 7-1c

BIOLOGY Biologists often examine organisms under microscopes 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. Example 7-2a

Draw a model using a circle 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 Example 7-2b

Take the square root of each side. Segment products Substitution Simplify. Take the square root of each side. Answer: 0.66 mm Example 7-2c

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 Example 7-2d

Find x if EF 10, EH 8, and FG 24. Example 7-3a

Secant Segment Products Substitution Distributive Property Subtract 64 from each side. Divide each side by 8. Answer: 34.5 Example 7-3b

Find x if and Answer: 26 Example 7-3c

Find x. Assume that segments that appear to be tangent are tangent. Disregard the negative solution. Answer: 8 Example 7-4a

Find x. Assume that segments that appear to be tangent are tangent. Answer: 30 Example 7-4c

End of Lesson 7

Example 1 Equation of a Circle Example 2 Use Characteristics of Circles Example 3 Graph a Circle Example 4 A Circle Through Three Points Lesson 8 Contents

Write an equation for a circle with the center at (3, –3), d 12. Equation of a circle Simplify. Answer: Example 8-1a

Write an equation for a circle with the center at (–12, –1), r 8. Equation of a circle Simplify. Answer: Example 8-1b

Write an equation for each circle. a. center at (0, –5), d 18 b. center at (7, 0), r 20 Answer: Answer: Example 8-1c

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. Sketch a drawing of the two tangent lines. Example 8-2a

Since d 10, r 5. The line x –1 is perpendicular to a radius 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. Example 8-2b

The center is at (4, 2), and the radius is 5. Likewise, the radius perpendicular to the line y –3 lies on a vertical line. The value of k is 5 units up from –3. The center is at (4, 2), and the radius is 5. Answer: An equation for the circle is . Example 8-2b

A circle with a diameter of 8 has its center in the second quadrant 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. Answer: Example 8-2c

Compare each expression in the equation to the standard form. Graph Compare 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. Example 8-3a

Answer: Example 8-3b

Write the expression in standard form. 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). Example 8-3c

Answer: Example 8-3d

a. Graph Answer: Example 8-3e

b. Graph Answer: Example 8-3f

Explore You are given three points that lie on a circle. 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. Example 8-4a

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. Example 8-4b

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: Example 8-4d

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: Example 8-4e

End of Lesson 8