UNIT 9 ircles Look around whatever room you are in and notice all the circular shapes. Perhaps you see a clock with a circular face, the rim of a cup or glass, or the top of a fishbowl. ircles have perfect symmetry, and their beauty can be seen in the design of many everyday objects. alph Waldo Emerson, in his essay called ircles, described them wonderfully: The eye is the first circle; the horizon which it forms is the second; and throughout nature this primary figure is repeated without end. ut circles are not used just for beauty; they are used for function as well. Imagine the problems we would have with square Ferris wheels or rectangular bicycle wheels. In this unit, you will study the parts of a circle and the relationships that eist among lines, angles, and arcs in circles. UNIT OJETIVE Identify parts of a circle. Find the degree measure of an arc and the length of an arc. olve problems involving inscribed angles, central angles, and intercepted arcs. Use theorems involving secants, tangents, chords, and arcs to solve problems. Use theorems involving secant segments, tangent segments, and chord segments to find missing measures. Write the equation of a circle and sketch a circle from its equation. unit 9 ILE 259 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
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hords and rcs Objectives efine and name a circle and the parts of a circle. Identify and define major arcs, minor arcs, and semicircles. Find the degree measure and length of an arc. Use the hords and rcs Theorem and its converse. Ferris wheel can help you understand the parts of a circle. Think of each car as a point on a circle. ll the cars are the same distance away from the center of the Ferris wheel. support beam that connects a car to the center is a radius. The car that is at the very bottom and the car that is at the very top are endpoints of a diameter. You will soon learn more circle terms that can be associated with the Ferris wheel. KEYWO arc arc length center central angle chord circle diameter inscribed angle major arc minor arc radius semicircle Parts of ircles emember N M ll radii of a circle have equal length. ll diameters of a circle have equal length. T P circle is the set of all points in a plane that are equidistant from a given point in the plane called the center. ircles are named by their center. The circle shown is circle M. radius of a circle is a segment that connects the center to a point on the circle. In circle M, MN, M, and MP are radii. diameter of a circle is a segment that connects two points on the circle and passes through the center. In circle M, NP is a diameter. chord connects any two points on the circle. In circle M, T and NP are chords. Notice that NP is also a diameter. iameters are chords that pass through the center of a circle. n arc is an unbroken part of a circle. It is formed by two points on the circle and the continuous part of the circle that lies between the two points. If the endpoints of an arc are also the endpoints of a diameter, the arc is a semicircle. hords and rcs 261 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
W X E Z T V Y rcs smaller than a semicircle are called minor arcs and are named by only their endpoints. y contrast, the names of semicircles and major arcs, which are arcs larger than a semicircle, include another point between the endpoints. In circle, is a minor arc. We could also name it. If we want the arc that has endpoints and but is not shaded red, we could write or E. Two semicircles in circle are and E. When we place chords and radii on a circle, we create angles. The type of angle formed depends on the location of the verte. If the verte is the center of the circle, we call the angle a central angle. If the verte is on the circle, we call it an inscribed angle. In circle T, XTW, WTZ, and ZTV are central angles and VXY is an inscribed angle. ngle and rc Measures The degree measure of an arc is the same as the degree measure of its corresponding central angle. In circle, m = 90, so m = 90 ince a complete rotation measures 360, m X = 360 m = 360 90 = 270 X rc Measures The degree measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360 minus the degree measure of its central angle. The degree measure of a semicircle is 180. Two circles are congruent if they have the same radius. Two arcs are said to be congruent if they are on congruent circles and have the same angle measure. The circumference of a complete circle is 2πr. ecause an arc is part of a circle, arc length is a part of the circumference of a circle. It can be found by writing the part as a fraction over 360 and multiplying by the circumference. Formula for rc Length The length L of an arc, where m is the degree measure of the arc and r is the radius, is L = ( m 360 ) 2πr 5 cm X To find the length of, L = ( 90 360 ) 2π(5) = ( 4 ) 10π = 2.5π 7.85 The arc length of is about 7.85 cm. 262 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
hords and rcs hords and arcs have the following relationship. Theorem 9-1 hords and rcs Theorem In a circle or in congruent circles, the arcs of congruent chords are congruent. Given ircle with F Prove F F tatement eason ketch 1. ircle with F Given F 2. F ll radii of a circle are congruent. F 3. F ongruence Postulate F 4. F PT F 5. F The measure of a minor arc is the measure of its central angle. F hords and rcs 263 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
The converse of the hords and rcs Theorem is also true. THEOEM 9-2 onverse of the hords and rcs Theorem In a circle or in congruent circles, the chords of congruent arcs are congruent. ummary circle is the set of all points in a plane that are a given distance from a center point. segment that connects the center to the circle is a radius. segment that connects two points on the circle is a chord. If the chord passes through the center, the chord is also a diameter. n arc is part of a circle. minor arc is less than half the circle, a major arc is more than half the circle, and a semicircle is half the circle. central angle of a circle is an angle whose verte is the center of the circle. n inscribed angle of a circle has its verte on the circle. The degree measure of a minor arc is the measure of its central angle. The degree measure of a major arc is the difference between the measure of the minor arc and 360. ll semicircles measure 180. The length L of an arc is a fraction of the circle s circumference, or L = ( m 360 ) 2πr, where r is the radius of the circle and m is the measure of the central angle. In a circle or in congruent circles, the arcs of congruent chords are congruent. lso, the chords of congruent arcs are congruent. 264 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
Tangents to ircles Objectives efine and identify tangents and secants. Use the Tangent Theorem and its converse. evelop and use the adius and hord Theorem. how that the perpendicular bisector of a chord passes through the center of a circle. Look at the train wheel on the track below. If you consider the wheel to be a circle and the track it rolls on to be a line, then you can say that the track is tangent to the train wheel. The wheel always touches the track at just one point. KEYWO point of tangency secant tangent Tangents and ecants E F tangent to a circle is a line in the plane of the circle that intersects the circle in eactly one point. In the diagram at left, is tangent to circle. You can also say is a tangent of circle. The point where the circle and the tangent intersect is called the point of tangency. Point is the point of tangency in this figure. secant is a line that intersects a circle in two points. EF is a secant of circle. Notice that secant EF contains chord EF. Tangent Theorems THEOEM 9-3 The Tangent Theorem line that is tangent to a circle is perpendicular to a radius of the circle at the point of tangency. Tangents to ircles 265 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
The sketch below shows that tangent is perpendicular to radius at the point of tangency,. 55 You can use the Tangent Theorem to find angle measures. In the diagram, is tangent to circle and m = 55. EONNET TO THE IG IE emember The sum of the measures of the angles of a triangle is 180. The Tangent Theorem tells us that m = 90. The sum of the measures of the angles of a triangle is 180 so, m + m + m = 180 90 + 55 + = 180 145 + = 180 = 35 The onverse of the Tangent Theorem is also true. THEOEM 9-4 The onverse of the Tangent Theorem line that is perpendicular to a radius of a circle at its endpoint on the circle is tangent to the circle. 8 15 17 EONNET TO THE IG IE emember The Pythagorean Theorem is a fundamental tool in geometry. You can use the onverse of the Tangent Theorem to determine if a line is tangent to a circle. To determine if is a tangent to circle, you use the onverse of the Pythagorean Theorem to see if is a right triangle. c 2 = a 2 + b 2 17 2 = 8 2 + 15 2 289 = 64 + 225 289 = 289 is a right triangle with right angle. o, and is tangent to circle. 266 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
adii and hords There is a relationship between the radius of a circle and a chord of that circle. Theorem 9-5 The adius and hord Theorem radius that is perpendicular to a chord of a circle bisects the chord. Given Prove ircle with radius P bisects P. P tatement eason ketch 1. ircle with radius P Given P 2. and P are right angles. efinition of perpendicular lines 3. raw P and. Two points determine a line. P P 4. P ll radii of a circle are congruent. 5. efleive Property of ongruence P P proof continued on net page Tangents to ircles 267 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
sesyyy tatement eason ketch 6. P HL ongruence Theorem P 7. P PT P 8. is the midpoint of P. efinition of midpoint 9. bisects P. efinition of a segment bisector P P The perpendicular bisector of any chord is a diameter of the circle and therefore will pass through the center of the circle. In the sketch shown, the perpendicular bisector E of chord was constructed. E passes through the center of the circle. ummary tangent to a circle is a line in the plane of the circle that intersects the circle in eactly one point. The point where the circle and the tangent intersect is called the point of tangency. secant is a line that intersects a circle in two points. Every secant contains a chord. The Tangent Theorem states that a line that is tangent to a circle is perpendicular to a radius of a circle at the point of tangency. The onverse of the Tangent Theorem is also true. radius that is perpendicular to a chord of a circle bisects the chord. The perpendicular bisector of a chord passes through the center of the circle because it is always a diameter of the circle. 268 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
Inscribed ngles and rcs Objectives efine and identify inscribed angles and their intercepted arcs. Use the Inscribed ngle Theorem and its corollaries to find measures of inscribed angles and intercepted arcs. tring art is created by hammering nails into a board in a geometric shape and wrapping and arranging string around the nails to create a symmetrical design. Though the string forms a series of line segments, our eyes see circular shapes. You begin creating the design that is shown by constructing sets of inscribed angles. These form the points around the circumference of the design. KEYWO inscribed angle intercepted arc Inscribed ngles and Intercepted rcs E 100 n inscribed angle of a circle has its verte on the circle and its sides are chords of the circle. E is an inscribed angle of the circle at left. We call the intercepted arc of E. We can also say that E intercepts. The endpoints of an intercepted arc are the points where the sides of the inscribed angle meet the circle. The net theorem describes how the measure of the inscribed angle relates to the measure of its intercepted arc. 50 THEOEM 9-6 Inscribed ngle Theorem n angle inscribed in a circle has a measure that equals one-half the measure of its intercepted arc. Inscribed ngles and rcs 269 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
There are three cases to consider when proving the theorem. ase 1: The center of the circle lies on a side of the inscribed angle. ase 2: The center of the circle lies inside the inscribed angle. ase 3: The center of the circle lies outside the inscribed angle. Here is the proof of ase 1. Given ircle with m = Prove m = 2 m tatement eason ketch 1. ircle with m = Given proof continued on net page 270 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
tatement eason ketch 2. raw. Two points determine a line. 3. ll radii of a circle are congruent. 4. is isosceles. efinition of isosceles triangle 5. Isosceles Triangle Theorem 6. m = 2 Eterior ngle Theorem 2 7. m = 2 The degree measure of a minor arc is the measure of its central angle. 2 2 8. m = 2 m ubstitution Property of Equality 2 2 Inscribed ngles and rcs 271 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
ecall that a semicircle measures 180. n inscribed angle that intercepts a semicircle also measures 90. This is the net corollary. OOLLY 9-1 ight-ngle orollary n angle that is inscribed in a semicircle is a right angle. Every arc has many inscribed angles. E OOLLY 9-2 rc-intercept orollary Two inscribed angles that intercept the same arc have the same measure. E and E both intercept E. o, m E = m E. ummary n inscribed angle of a circle has its verte on the circle and its sides are chords of the circle. The arc opposite the angle is the intercepted arc. The endpoints of the intercepted arc are the points where the sides of the inscribed angle meet the circle. The measure of an inscribed angle equals one-half the measure of its intercepted arc. n angle that is inscribed in a semicircle is a right angle. Two inscribed angles that intercept the same arc have the same measure. 272 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
ngles Formed by ecants and Tangents Objectives Find the measures of angles formed by intersecting tangents and secants. Find the measures of intercepted arcs formed by intersecting tangents and secants. solar eclipse occurs when the moon is between the earth and the sun at a position that causes the moon s shadow to fall on the earth s surface. Though it may seem like this would happen frequently, it is a rather rare occurrence. The moon s orbit around the earth is tilted by 5 to the earth s orbit around the sun. The tilting causes the moon s shadow to almost always miss the earth s surface. The lines in the diagram show what happens when the moon completely blocks the rays of the sun from reaching a particular spot on the earth. The lines are tangent to the circles representing the sun and moon. sun moon earth More ngle and rc elationships o far you have learned the relationships between angles and arcs for both central and inscribed angles. Now you will learn even more angle and arc relationships. They fall into three basic categories. 1. The verte is on the circle. 2. The verte is outside the circle. 3. The verte is inside the circle. Verte on the ircle The Inscribed ngle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. It turns out that this theorem can be etended to include angles other than inscribed angles whose verte is on a circle. This happens when a tangent and secant intersect. THEOEM 9-7 If a tangent and a secant or chord intersect on a circle at the point of tangency, then the measure of the angle formed equals one-half the measure of the intercepted arc. ngles Formed by ecants and Tangents 273 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
130 G m G = 2 m G = 2 (130 ) = 65 Verte Outside the ircle There are three ways for two lines to intersect so that the verte is outside the circle. In each of these cases, there are two intercepted arcs. lso, in each, the measure of the angle is half the difference of the intercepted arcs. These are our net three theorems. Theorem 9-8 The measure of a secant-tangent angle with its verte outside the circle equals one-half of the difference of the measures of the intercepted arcs. Given is a tangent. is a secant. Prove m = 2 ( m m E ) E tatement eason ketch 1. is a tangent. is a secant. Given E 2. raw. Two points determine a line. E 3. m = 2 m E Inscribed ngle m = 2 m Theorem 4. m + m = m Eterior ngle Theorem 5. 2 m E + m = m ubstitution Property of 2 Equality 6. m = 2 m 2 m E ubtraction Property of Equality 7. m = 2 ( m m E ) istributive Property 274 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
50 We will use the theorem to find m. m = 2 (220 50 ) = 2 (170 ) = 85 220 THEOEM 9-9 The measure of an angle that is formed by two secants that intersect in the eterior of a circle equals one-half of the difference of the measures of the intercepted arcs. 120 W 50 P m W = 2 (m W m P) = 2 (120 50 ) = 2 (70 ) = 35 THEOEM 9-10 238 The measure of a tangent-tangent angle with its verte outside the circle equals one-half of the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180. T 122 m T = 2 (m T m T) = 2 (238 122 ) = 2 (116 ) = 58 Verte Inside the ircle The only way for two lines to intersect inside a circle is if both lines are secants. THEOEM 9-11 W P 105 75 T The measure of an angle that is formed by two secants or chords that intersect in the interior of a circle equals one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. m P = 2 (m P + m WT) = 2 (105 + 75 ) = 2 (180 ) = 90 ngles Formed by ecants and Tangents 275 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
These theorems can be used to find missing arcs as well as missing angles. It just takes a little algebra. 66 W 52 T m W = 2 (m + m T) 66 = 2 ( + 52 ) 132 = + 52 80 = ummary ecants and tangent lines of circles form angles that intercept arcs. The verte can lie on, outside, or inside the circle. When the verte lies on the circle, there is one intercepted arc, and the measure of the angle is half the measure of the intercepted arc. When the verte lies outside the circle, there are two intercepted arcs, and the measure of the angle is half the difference of the measures of the intercepted arcs. When the verte lies inside the circle, there are two intercepted arcs, and the measure of the angle is half the sum of the measures of the intercepted arcs. 276 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
egments of Tangents, ecants, and hords Objectives efine and identify tangent segments, secant segments, and eternal secant segments. Use theorems about tangents, secants, and chords to find missing segment lengths. One of the oldest types of bridges in the world is the arch bridge. oman builders were among the first to build arch bridges. Whereas modern builders use other types of arcs, the omans used semicircles. You can find the diameter of the circle that contains the arc with what you will learn in this lesson. KEYWO eternal secant segment secant segment tangent segment Tangent egments tangent segment is a segment of a tangent line with one of its endpoints at the point of tangency. In the diagram, and are tangent segments. egments of Tangents, ecants, and hords 277 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
Theorem 9-12 Two segments that are tangent to a circle from the same eternal point are of equal length. Given Prove YX and YZ are tangent segments of circle W. YX YZ Y X Z W tatement eason ketch 1. YX and YZ are tangent segments. Given X W 2. raw XW, YW, and ZW. Two points determine a line. Y X Z W Y Z 3. YXW and YZW are right angles. tangent is perpendicular to a radius at the point of tangency. Y X W Z 4. WX WZ ll radii of a circle are congruent. X W Y Z 5. YW YW efleive Property of ongruence X W Y Z 6. YXW and YZW are right triangles. ee tep 3. efinition of right triangles X W Y Z 7. YXW YZW HL ongruence Theorem X W Y Z proof continued on net page 278 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
tatement eason ketch 8. YX YZ PT X W Y Z ecant egments secant segment is a segment of a secant with at least one of its endpoints on the circle. elow, N, JN, TN, and MN are secant segments. J N M T n eternal secant segment is a secant segment that lies in the eterior of the circle with one of its endpoints on the circle. JN and MN are eternal secant segments. The net theorem gives the relationship among these segments. THEOEM 9-13 If two secants intersect outside a circle, then the product of the lengths of one secant segment and its eternal segment equals the product of the lengths of the other secant segment and its eternal segment. (Whole Outside = Whole Outside) T T = T MT T M You can use this theorem to find missing segment lengths like MK at left. G 8 M H K 7 6 J GJ HJ = MJ KJ (8 + 6)(6) = ( + 7)(7) (14)(6) = ( + 7)(7) 84 = 7 + 49 35 = 7 5 = MK = 5 egments of Tangents, ecants, and hords 279 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
ecant-tangent egments E THEOEM 9-14 If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its eternal segment equals the length of the tangent segment squared. (Whole Outside = Tangent 2 ) = E 2 5 cm G 4 cm Use Theorem 9-14 to find G. = G 2 (5 + 4)(4) = 2 9(4) = 2 36 = 2 6 = G = 6 cm hord-hord egments THEOEM 9-15 If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. = T T 5 m 10 m E 6 m Find E. E E = E E (5)(6) = (10) 30 = 10 3 = E = 3 m 280 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
35 ft 120 ft 35 60 60 n arch of a bridge that passes over a river is the arc of a circle. s shown at left, the height from the center of the arc to the water below the bridge is 35 feet. The distance from one side of the bridge to the other, at the water line, is 120 feet. Find the diameter of the circle that includes the arc. The situation can be viewed as a 120-ft chord being bisected by a perpendicular segment from the midpoint of the arc. First find the missing distance,. 35 = 3600 102.9 To find the diameter, add 35: 102.9 + 35 = 137.9. The diameter is about 138 feet. ummary tangent segment is a segment of a tangent line with one of its endpoints at the point of tangency. secant segment is a segment of a secant with at least one of its endpoints on the circle. n eternal secant segment is a secant segment that lies in the eterior of the circle with one of its endpoints on the circle. egments that are tangents to a circle from the same eternal point are equal in length. If two secants intersect outside the circle, the length of a secant segment (whole) times the length of its eternal segment (outside) equals the length of the other secant segment (whole) times its eternal segment (outside). If a secant and tangent intersect outside the circle, the length of the secant segment (whole) times the length of the eternal secant segment (outside) equals the length of the tangent segment squared. If two chords intersect, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. egments of Tangents, ecants, and hords 281 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
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ircles in the oordinate Plane Objectives Write the equation of a circle given its center and radius, or its center and another point on the circle. ketch a circle given its equation. eismologists use intersecting circles to determine the epicenter of an earthquake. The point underground where the earthquake releases energy is called the focus of the earthquake. The point directly above the focus on the earth s surface is called the epicenter. uring an earthquake, seismologists use a seismograph to measure vibrations in three different locations. Then they draw a circle around each location on a map. The radius of each circle is the distance of the earthquake from the seismograph. The intersection of the three circles is the epicenter. Find the intercepts of a circle. Translate a circle on the coordinate plane. Minneapolis etroit approimate epicenter harleston The Equation of a ircle emember The distance between two points ( 1, y 1 ) and ( 2, y 2 ) is d = ( 2 1 ) 2 + (y 2 y 1 ) 2. You can use the istance Formula to understand the formula for the equation of a circle. The diagram shows a circle with center (h, k) and radius r. y (, y) r (h, k) O For any point (, y) on the circle, you represent the distance r from the fied point (h, k) by using the istance Formula: r = ( h) 2 + (y k) 2 quaring each side, we get the following: r 2 = ( h) 2 + (y k) 2 ircles in the oordinate Plane 283 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
The Equation of a ircle The equation of a circle with center (h, k) and radius r is ( h) 2 + (y k) 2 = r 2 y (h, k) r (, y) If you know the center and the radius of a circle, you can write the equation of the circle. For eample, to find the equation of a circle with center (3, 5) and radius 4, substitute those values into the formula: O ( h) 2 + (y k) 2 = r 2 ( 3) 2 + (y 5) 2 = 4 2 ( 3) 2 + (y 5) 2 = 16 Notice that the operation in each set of parentheses in the formula is subtraction. emember that subtracting a negative is the same as adding a positive. For eample, the equation of a circle with center ( 1, 0) and radius 3 is ( ( 1)) 2 + (y 0) 2 = 3 2 ( + 1) 2 + y 2 = 9 If you know the center of the circle and a point on the circle, you can find the radius and then write the equation of the circle. MemoryTip Just as precedes y in the alphabet, h precedes k. o h goes with and k goes with y. To find the equation of a circle with center (3, 2) that passes through the point (7, 5), you first find the radius by using the istance Formula. r = (3 7) 2 + ( 2 ( 5)) 2 = ( 4) 2 + ( 2 + 5) 2 = ( 4) 2 + (3) 2 = 25 = 5 Now substitute 5 for r, 3 for h, and 2 for k in the equation and simplify. ( 3) 2 + (y ( 2)) 2 = 5 2 ( 3) 2 + (y + 2) 2 = 25 284 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
ketching a ircle If you know the equation of a circle, you know its radius and its center. You can use this information to sketch the circle. Let s sketch the circle described by the equation: ( 2) 2 + (y) 2 = 16 y 5 4 3 2 1-3 -2-1 1 2 3 4-1 -2 (2, 0) -3-4 -5 5 6 7 The equation can be rewritten as ( 2) 2 + (y 0) 2 = 4 2, where (2, 0) is the center of the circle and 4 is the radius. tep 1: Plot the center point (2, 0). tep 2: From the center point, count the number of units of the radius, 4, up, down, left, and right and place a point at each location. tep 3: ketch the circle that passes through these four points. compass would help with this sketch. emember ecause we are looking for coordinates, and not distances, we keep both the positive and negative roots. You can see from the sketch that the -intercepts are 2 and 6 and that the y-intercepts are approimately 3.5 and 3.5. To find the intercepts algebraically, substitute 0 for y to find the -intercepts and 0 for to find the y-intercepts. -intercepts y-intercepts ( 2) 2 + ( y) 2 = 16 ( 2) 2 + ( y) 2 = 16 ( 2) 2 + (0) 2 = 16 (0 2) 2 + ( y) 2 = 16 ( 2) 2 = 16 4 + y 2 = 16 2 = ±4 y 2 = 12 = 6 and = 2 y = ± 12 y 3.46 and y 3.46 The -intercepts are (6, 0) and ( 2, 0). The y-intercepts are approimately (0, 3.46) and (0, 3.46). You can also use geometry software to sketch a circle on the coordinate plane. The sketch below is a graph of the circle ( 4) 2 + (y 2) 2 = 9. ircles in the oordinate Plane 285 opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.
emember The coordinates of the center point follow the minus sign. Use parentheses when translating to avoid errors with the signs. You can translate a circle by moving its center. To move the previous circle 2 units to the right and 3 units up, add 2 to h and 3 to k. The image circle will have the equation ( (4 + 2)) 2 + (y (2 + 3)) 2 = 9, which simplifies to ( 6) 2 + (y 5) 2 = 9. You can also translate a circle with geometry software. raw a vector from the center of the circle to a point that is 2 units to the right and 3 units up. Then translate the circle by the vector. ummary The equation of a circle with center (h, k) and radius r is ( h) 2 + (y k) 2 = r 2. If you know the center and the radius of a circle, you can write the equation of the circle. If you know the center of the circle and a point on the circle, you can find the equation of the circle by first using the istance Formula to find the radius. To sketch a circle in the coordinate plane, plot the center point, and then count the number of units of the radius in each direction to plot four other points to guide you in sketching the circle. To find the - and y-intercepts algebraically, substitute 0 for y and solve for, then substitute 0 for and solve for y. You can translate a circle by adding and subtracting units from the coordinates of the center point. 286 Unit 9 ILE opyright 2006, K12 Inc. ll rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K12 Inc.