12-1. Tangent Lines. Vocabulary. Review. Vocabulary Builder HSM11_GEMC_1201_T Use Your Vocabulary

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1 1-1 Tangent Lines Vocabulary Review 1. ross out the word that does NT apply to a circle. arc circumference diameter equilateral radius. ircle the word for a segment with one endpoint at the center of a circle and the other endpoint on the circle. arc circumference diameter perimeter radius Vocabulary uilder tangent (noun, adjective) tan tangent junt efinition: tangent to a circle is a line, ray, or segment in the plane of the circle that intersects the circle in exactly one point. * ) Examples: In the diagram, is tangent to the circle at. is the point of tangency. ) is a tangent ray. is a tangent segment. HSM11_GEM_11_T9313 ther Usage: In a right triangle, the tangent is the ratio of the side opposite an acute angle to the side adjacent to the angle. Use Your Vocabulary 3. omplete each statement with always, sometimes, or never. diameter is 9 a tangent. never tangent and a circle 9 have exactly one point in common. always radius can 9 be drawn to the point of tangency. always tangent 9 passes through the center of a circle. never tangent is 9 a ray. sometimes hapter ther Word orm: tangency (noun)

2 Theorems 1-1, 1-, and 1-3 Theorem 1-1 If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency. Theorem 1- If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. Theorem 1-3 If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. Use the diagram at the right for Exercises 4 6. omplete each statement. * ) * ) 4. Theorem 1-1 If * ) is tangent to ( at K, then ' K. * ) 5. Theorem 1- If ' K, then is tangent to (. K 6. Theorem 1-3 If and are tangent to (, then >. Problem 1 inding ngle Measures Got It? E is tangent to (. What is the value of x? 7. ircle the word that best describes. diameter radius tangent 38 x E 8. What relationship does Theorem 1-1 support? ircle your answer. ' E 6 E > E 9. ircle the most accurate description of the triangle. acute isosceles obtuse right 1. ircle the theorem that you will use to solve for x. Theorem omplete the model below. Relate Write 1. Solve for x. sum of angle measures in a triangle x x x x Triangle ngle-sum Theorem measure is 38 plus plus of measure of E 18 9 x = The value of x is Lesson 1-1

3 Problem inding istance Got It? What is the distance to the horizon that a person can see on a clear day from an airplane mi above Earth? Earth s radius is about 4 mi. 14. The diagram at the right shows the airplane at point and the horizon at point H. Use the information in the problem to label the distances. 15. Use the justifications at the right to find the distance. H ' H Theorem 1-1 H mi 4 mi H 1 H 5 Pythagorean Theorem 4 1 H 5 4 Substitute. 16,, 1 H 5 16,16,4 Use a calculator. 4 mi H 5 16,4 Subtract from each side. H 5 Å 16,4 Take the positive square root. H < Use a calculator. 16. person can see about 17 miles to the horizon from an airplane mi above Earth. Problem 3 inding a Radius Got It? What is the radius of (? 17. Write an algebraic or numerical expression for each side of the triangle. x x 1 6 x 18. ircle the longest side of the triangle. Underline the side that is opposite the right angle. 1 x x Use the Pythagorean Theorem to complete the equation. x ( x 1 6 ). Solve the equation for x. 1 x (x 1 6) x x 1 1x 1 36 x 1 1 x 5 x 1 1x 1 36 x 1 5 1x x x x x The radius is hapter 1 316

4 Problem 5 ircles Inscribed in Polygons 15 cm Got It? ( is inscribed in kpqr, which has a perimeter of 88 cm. What is the Q X P length of QY? Z PZ, RZ > RY, and QX > QY, so PX 5 PZ,. y Theorem 1-3, PX > Y 17 cm R RY, and QX 5 QY. RZ 5 XQ 1 QY 1 YR 1 RZ 1 ZP 3. Perimeter p 5 PQ 1 QR 1 RP, so p 5 PX 1 by the Segment ddition Postulate. hsm11gmse_11_t5 4. Use the values in the diagram and your answer to Exercise 3 to solve for QY QX 1 QY QY 5 QY 5 QY Lesson heck o you UNERSTN? Error nalysis classmate insists that is a tangent to (E. Explain how to show that your classmate is wrong. E Underline the correct word or number to complete the sentence tangent to a circle is parallel / perpendicular to a radius If is tangent to (E at point, then m/e must be 3 / 9 / triangle can have at most 1 right angle(s). 8. Explain why your classmate is wrong. nswers may vary. Sample: hsm11gmse_11_t777 triangle cannot have two right angles, so the segment is not a tangent. Math Success heck off the vocabulary words that you understand. circle tangent to a circle point of tangency Rate how well you can use tangents to find missing lengths. Need to review Now I get it! Lesson 1-1

5 1- hords and rcs Vocabulary Review ircle the converse of each statement. 1. Statement: If I am happy, then I sing. If I sing, then I am happy. If I am not happy, then I do not sing. If I do not sing, then I am not happy.. Statement: If parallel lines are cut by a transversal, then alternate interior angles are congruent. If lines cut by a transversal are not parallel, then alternate interior angles are not congruent. If lines cut by a transversal If lines cut by a transversal form form alternate interior alternate interior angles that are angles that are not congruent, then the lines are congruent, then the parallel. lines are not parallel. chord (noun) kawrd chord RS efinition: chord is a segment whose endpoints are on a circle. S R Related Word: arc Use Your Vocabulary 3. omplete each statement with always, sometimes, or never. HSM11_GEM_1_T93876 chord is 9 a diameter. sometimes diameter is 9 a chord. always radius is 9 a chord. never chord 9 has a related arc. always n arc is 9 a semicircle. sometimes hapter Vocabulary uilder

6 Theorems 1-4, 1-5, 1-6 and Their onverses Theorem 1-4 Within a circle or in congruent circles, congruent central angles have congruent arcs. 4. If / > l, then >. onverse Within a circle or in congruent circles, congruent arcs have congruent central angles. 5. If >, then / > l. Theorem 1-5 Within a circle or in congruent circles, congruent central angles have congruent chords. 6. If / > /, then >. onverse Within a circle or in congruent circles, congruent chords have congruent central angles. 7. If >, then l > /. Theorem 1-6 Within a circle or in congruent circles, congruent chords have congruent arcs. 8. If >, then >. onverse Within a circle or in congruent circles, congruent arcs have congruent chords. 9. If >, then >. Problem 1 P Given Using ongruent hords Got It? Use the diagram at the right. Suppose you are given ( (P and l lp. How can you show l lp? rom this, what else can you conclude? 1. omplete the flow chart below to explain your conclusions. P Isosceles Theorem P circles have radii. and Given Transitive Property S P P P orresponding parts of s are. Theorem 1-4 Theorem Lesson 1-

7 Theorem 1-7 and Its onverse, Theorems 1-8, 1-9, 1-1 Theorem 1-7 Within a circle or in congruent circles, chords equidistant from the center or centers are congruent. onverse Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers). 11. If E 5, then >. 1. If >, then E 5. Theorem 1-8 In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. 13. If is a diameter and ', then E > E and >. Theorem 1-9 In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord. 14. If is a diameter and E > E, then '. Theorem 1-1 In a circle, the perpendicular bisector of a chord contains the center of the circle. 15. If is the perpendicular bisector of chord, then contains the center of (. E E Problem inding the Length of a hord Got It? What is the value of x? Justify your answer. 16. What is the measure of each chord? Explain. 36. Explanations may vary. Sample: The length of one chord is 36 and the other chord has two segments with lengths 18, so, by the Segment ddition Postulate, it also has a length of ircle the reason why the chords are congruent. hords that have equal hords that are equidistant from measures are congruent. the center of a circle are congruent. 18. ircle the theorem that you will use to find the value of x. Theorem 1-5 Theorem 1-7 onverse of Theorem 1-7 Theorem 1-8 Theorem ircle the distances from the center of a circle to the chords x x. The value of x is 16. hapter 1 3

8 Problem 3 Using iameters and hords Got It? The diagram shows the tracing of a quarter. What is its radius? Underline the correct word to complete each sentence. Then do each step. 1. irst draw two chords / tangents.. Next construct one / two perpendicular bisector(s). 3. Label the intersection. It is the circle s center / chord. 4. Measure the diameter / radius. 5. The radius is about 1 mm. Problem 4 inding Measures in a ircle Got It? Reasoning In finding y, how does the auxiliary make the problem simpler to solve? 6. is the hypotenuse of a right 9, so you can use the 9 Theorem to solve for y. triangle Pythagorean y E 15 Lesson heck o you UNERSTN? Vocabulary Is a radius a chord? Is a diameter a chord? Explain your answers. 7. ircle the name(s) of figure(s) that have two endpoints on a circle. Underline the name(s) of figure(s) that have one endpoint on a circle. chord diameter radius ray segment 8. Is a radius a chord? Is a diameter a chord? Explain. nswers may vary. Sample: radius has only one endpoint on a circle, so it is not a chord. diameter has two endpoints on a circle, so it is a chord. Math Success heck off the vocabulary words that you understand. circle chord radius diameter Rate how well you can use chords to find measures. Need to review Now I get it! 31 Lesson 1-

9 1-3 Inscribed ngles Vocabulary Review Write noun or verb to identify how intercept is used. 1. efense tries to intercept a touchdown pass. verb. The y-intercept of a line is the y-value at x 5. noun 3. ryptographers intercept and decipher code messages. verb 4. The x-intercept of a line is the x-value at y 5. noun Vocabulary uilder Related Word: circumscribed efinition: Inscribed means written, marked, or engraved on. ircumscribed means encircled, confined, or limited. Math Usage: n inscribed angle is formed by two chords with a vertex on the circle. HSM11_GEM_13_T93878 Use Your Vocabulary Write circumscribed or inscribed to describe each angle. 5. inscribed circumscribed circumscribed 8. inscribed HSM11_GEM_13_T9388 Underline the correct word to complete each sentence. HSM11_GEM_13_T9388 HSM11_GEM_13_T93881 HSM11_GEM_13_T / with points,, and on a circle is a(n) circumscribed / inscribed angle. hapter 1 1. n intercepted arc is between the sides of a(n) circumscribed / inscribed angle. 3 inscribed angle inscribed (adjective) in skrybd

10 Theorem 1-11 Inscribed ngle Theorem m/ 5 1 m The measure of an inscribed angle is half the measure of its intercepted arc. 11. Suppose m m/ 5? m 5 1. Suppose m/ 5 6.? m/ 5 1 m 5 hsm11gemc_13_t687 Problem 1 Using the Inscribed ngle Theorem Got It? In (, what is ml? 13. omplete the reasoning model below. Write Think I know the sides of are chords and the vertex is on. I can use the Inscribed ngle Theorem. is an inscribed angle. m = 1 (measure of the blue arc) 1 = 18 hsm11gmse_13_t11795.ai = 9 orollaries to Theorem 1-11 Inscribed ngle Theorem HSM11_GEM_13_T93883 orollary orollary 1 Two inscribed angles that intercept the same arc are congruent. orollary 3 n angle inscribed in a semicircle is a right angle. The opposite angles of a quadrilateral inscribed in a circle are supplementary. Use the diagram at the right. Write T for true or for false. T S 14. /P and /Q intercept the same arc. hsm11gmse_13_t856.ai hsm11gmse_13_t854.ai hsm11gmse_13_t855.ai 15. /SRP and /Q intercept the same arc. T T 16. TSR is a semicircle. 17. /PTS and /SRQ are opposite angles. T 18. /PTS and /SRP are supplementary angles. P 33 R Q HSM11_GEM_13_T93884 Lesson 1-3

11 Problem Using orollaries to ind ngle Measures Got It? In the diagram at the right, what is the measure of each numbered angle? 19. Use the justifications at the right to complete each statement. m/4 5 1 ( ) Inscribed ngle Theorem m/4 5 1 ( 14 ) dd within parentheses. m/4 5 7 Simplify.. ircle the corollary you can use to find m/. n angle inscribed in a semicircle The opposite angles of a quadrilateral is a right angle. inscribed in a circle are supplementary. 1. Now solve for m/. ml4 1 ml ml ml ml Underline the correct word to complete the sentence. The dashed line is a diameter / radius. 3. ircle the corollary you can use to find m/1 and m/3. n angle inscribed in a semicircle The opposite angles of a quadrilateral is a right angle. inscribed in a circle are supplementary. Use your answer to Exercise 3 to find the angle measures. 4. m/ m/ So, m/1 5 9, m/ 5 11, m/3 5 9, and m/ Theorem 1-1 The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. m/ 5 1 m 7. Suppose m/ Suppose m m 5? m/ 5 1 m/ 5? m 5 ) 9. In the diagram at the right, is tangent to ( at. m 5 18 m/ hapter 1 34

12 Problem 3 inding rc Measure Got It? In the diagram at the right, KJ is tangent to (. What are the values of x and y? 3. ircle the arc intercepted by /JQL. Underline the arc intercepted by /KJL. JL JQ QL QLJ 31. y the Inscribed ngle Theorem, mjl 5? y Theorem 1-1, x 5? m JL Q 35 y L J x K 33. The value of x is Underline the correct words to complete the sentence. QL is a diameter / radius, so /QJL is a(n) acute / right / obtuse angle. 35. Use the justifications at the right to complete each statement. m/qjl 1 m/jlq 1 m/lqj 5 18 Triangle ngle-sum Theorem 9 1 y Substitute. y Simplify. y 5 55 Subtract from each side. Lesson heck o you UNERSTN? Error nalysis classmate says that ml 5 9. What is your classmate s error? 36. Is diameter a side of /? Yes / No 37. Is / inscribed in a semicircle? Yes / No 38. What is your classmate s error? Explain. nswers may vary. Sample: The error is in thinking that an angle with a diameter as a side intercepts a semicircle. ml u 1? 18. Math Success heck off the vocabulary words that you understand. inscribed angle Need to review intercepted arc Rate how well you can find the measure of inscribed angles. Now I get it! 35 Lesson 1-3

13 ngle Measures and Segment Lengths 1-4 Vocabulary Review 1. Underline the correct word(s) to complete the sentence. The student went off on a tangent when he did / did not stick to the subject.. tangent to a circle intersects the circle at exactly 9 point(s). one 3. rom a point outside a circle, there are 9 tangent(s) to the circle. two Vocabulary uilder secant secant (noun) seek unt Related Word: tangent (noun) Source: The word secant comes from the Latin verb secare, which means to cut. * ) ) ) Examples: In the diagram at the right, is a secant, and are secant rays, HSM11_GEM_14_T93885 and is a secant segment. Use Your Vocabulary Write secant or tangent to identify each line Y X secant tangent tangent 7. X Y secant 8. Is a chord a secant? Explain. HSM11_GEM_14_T9385 HSM11_GEM_14_T9387 HSM11_GEM_14_T9384 HSM11_GEM_14_T9386 No. Explanations may vary. Sample: secant contains points outside a circle, while a chord does not. hapter 1 36 efinition: secant is a line that intersects a circle at two points.

14 Theorems 1-13, 1-14, and 1-15 m/1 5 1(x 1 y) Theorem 1-13 The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. x y 1 9. In the diagram at the right, does m/ 5 1(x 1 y)? Explain. Yes. Explanations may vary. Sample: l1 and l are vertical angles, HSM11_GEM_14_T93886 so ml1 5 ml and ml 5 1 (x 1 y) by the Trans. Prop. of Equality. Theorem 1-14 The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. m/1 5 1(x y) x y 1 y x 1 1. In the first diagam, the sides of the angle are a secant and a In the second diagram, the sides of the angle are a secant and a 9. hsm11gmse_14_t6969.ai 1. In the third diagram, the sides of the angle are a tangent and a Is m/1 5 (y x) equivalent to m/1 5 (x y)? x y 1 secant tangent tangent Yes / No Theorem 1-15 or a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle. I. II. a x c d P III. w b y t P y z P z omplete each case of Theorem ase I a? b 5 c? d hsm11gmse_14_t698 y 1 hsm11gmse_14_t ase II (w 1 x) w 5 ( z) y 16. ase III (y 1 z) y hsm11gmse_14_t t Problem 1 inding ngle Measures w Got It? What is the value of w? 17. Use Theorem 1-14 to complete the equation (w 11 ) 37 7 Lesson 1-4 hsm11gmse_14_t6976

15 18. Now solve the equation. 19. The value of w is (w 11) 14 5 w w Problem inding an rc Measure Got It? departing space probe sends back a picture of Earth as it crosses Earth s equator. The angle formed by the two tangents to the equator is. What arc of the equator is visible to the space probe?. Use,, G, and the words Earth and probe to complete the diagram below. Earth E P probe G 1. omplete the flow chart below. Let mg = x. Then meg = 36 x. 1 = ( x ) Substitute. 16 = 1 x Subtract 18 from each side. 16 = x ivide each side by 1. The sum of the arc measures is 36. m PG 1 = (meg m G ) Theorem = ( 36 x ) Simplify. = 18 x Use the istributive Property.. 16 arc of the equator is visible to the space probe. hapter 1 38

16 Problem 3 inding Segment Lengths 14 Got It? What is the value of the variable to the nearest tenth? Underline the correct word to complete each sentence. 16 x 3. The segments intersect inside / outside the circle. 4. Write a justification for each statement. (14 1 )14 5 (16 1 x)16 Use Theorem 1 15 (II) x Simplify. 5 16x Subtract 56 from each side. hsm11gmse_14_t6988 ivide each side by x 5. To the nearest tenth, the value of x is Lesson heck o you UNERSTN? In the diagram at the right, is it possible to find the measures of the unmarked arcs? Explain. z 6. You can use intercepted arcs to find the value of y. Yes / No 7. You can use supplementary angles to find the measures of the angles adjacent to y. Yes / No 8. You can find the sum of the unmarked arcs. Yes / No y 9 x Is it possible to find the measure of each unmarked arc? Explain. Explanations may vary. Sample: No. The sum of the unmarked angles is not enough information tohsm11gmse_14_t8585.ai find the measure of each unmarked arc. Math Success heck off the vocabulary words that you understand. chord circle secant tangent Rate how well you can find the lengths of segments associated with circles. Need to review Now I get it! 1 39 Lesson 1-4

17 1-5 ircles in the oordinate Plane Vocabulary Review T 1. The coordinate plane extends without end and has no thickness.. nly lines can be graphed in the coordinate plane. T 3. ny polygon can be plotted in the coordinate plane. 4. (, 5) and (5, ) are the same point in the coordinate plane. 5. The coordinate plane is three-dimensional. T 6. You can find the slope of a line in the coordinate plane. Vocabulary uilder standard form (noun) stan durd fawrm Main Idea: The standard form of an equation gives information that can help you graph the equation in the coordinate plane. Examples: The standard form of an equation of a circle is (x h) 1 (y k) 5 r. The standard form of a linear equation is x 1 y 5. The standard form of a quadratic equation is y 5 ax 1 bx 1 c. Use Your Vocabulary raw a line from each equation in olumn to its standard form in olumn. olumn 7. y 5 x y 5 4x 9. y 5 x 1. 5 y 4x 1 3 hapter 1 olumn x1y5 x y 5 3 3x 4y 5 8 4x y Write T for true or for false.

18 Theorem 1-16 Equation of a ircle n equation of a circle with center (h, k) and radius r is (x h) 1 (y k) 5 r. omplete each sentence with center, circle, or radius. 11. Each point on a 9 is the same distance from the center. circle y r (h, k) (x, y) 1. The equation of a circle with center (1, ) and 9 6 is (x 1 1) 1 (y ) 5 6. radius x 13. Each point on a circle is r units from the 9. center 14. The istance ormula is d 5 Å (x x ) 1 (y ) 1 y How is d in the istance ormula related to the radius r in the standard equation of a circle? nswers may vary. Sample: The radius r is the distance d between center (h, k) and (x, y) on a circle and is also the hypotenuse of a right triangle with vertices (h, k), (x, y), and (x h, y k). 16. How are the istance ormula and the standard form of the equation of a circle alike? nswers may vary. Sample: The standard form of the equation is the istance ormula using (x, y) on a circle and the center (h, k). This distance is the radius r of the circle. Problem 1 Writing the Equation of a ircle Got It? What is the standard equation of the circle with center (3, 5) and radius 6? 17. The x-coordinate of the center is The y-coordinate of the center is Is the standard equation of a circle (x h) 1 (y k) 5 d Yes / No. Identify the values of h, k, and r. h 5 3 k 5 5 r Write the standard equation of the circle with center (3, 5) and radius 6. (x 3 ) 1 ( y 5 ) 5 6. Simplify the equation in Exercise 1. (x 3 ) 1 ( y 5 ) Lesson 1-5

19 Problem Using the enter and a Point on a ircle Got It? What is the standard equation of the circle with center (4, 3) that passes through the point (1, 1)? 3. omplete the reasoning model below. Know Need Plan (h, k) is ( 4, 3 ). The radius r Use the istance ormula (1, 1) is a point on the The standard equation of to find r. circle. the circle Then substitute for (h, k) and for r. 4. Use the istance ormula to find r. d 5 Å (x x ) 1 (y ) 1 y 1 Write the istance ormula. r 5 Å (4 1 ) 1 (3 1 ) Substitute. r 5 Å ( 5 ) 1 ( ) Simplify within parentheses. r 5 Å ( 5 ) 1 ( 4 ) Square each number. r 5 Å 9 dd. 5. Now write the standard form of the circle with center (4, 3) that passes through the point (1, 1). (x h ) 1 ( y k ) 5 r Use the standard form of an equation of a circle. (x 4 ) 1 ( y 3 ) 5 "9 Substitute. (x 4 ) 1 ( y 3 ) 5 9 Simplify. Problem 3 Graphing a ircle Given Its Equation Got It? Suppose the equation (x 7) 1 (y 1 ) 5 64 represents the position and transmission range of a cell tower. What does the center of the circle represent? What does the radius represent? Place a in the box if the response is correct. Place an if it is incorrect. 6. The transmission range is the same distance all around the cell tower. 7. The center of the circle represents the position of the cell tower. 8. The center of the circle represents the transmission range. 9. The radius of the circle represents the position of the cell tower. 3. The radius of the circle represents the transmission range. hapter 1 33

20 Got It? What is the center and radius of the circle with 14 equation (x ) 1 (y 3) 5 1? Graph the circle. y The center of the circle is (, 3 ) r The radius of the circle is Graph the circle on the coordinate plane at the right x Lesson heck o you UNERSTN? HSM11_GEM_15_T93138 Suppose you know the center of a circle and a point on the circle. How do you determine the equation of the circle? 35. o you know the value of h? Yes / No 36. o you know the value of k? Yes / No 37. o you know the value of r? Yes / No 38. How can you find the missing value? nswers may vary. Sample: Use the istance ormula to find the radius r of the circle. 39. nce you know h, k, and r, how do you determine an equation of the circle? nswers may vary. Sample: Substitute the values in the standard equation of a circle. Math Success heck off the vocabulary words that you understand. circle istance ormula standard form Rate how well you can use the standard form of a circle. Need to review Now I get it! Lesson 1-5

21 1-6 Locus: Set of Points Vocabulary Review Write plane or space to complete each sentence. 1. The set of all points in three dimensions is 9. space. 9 is represented by a flat surface that extends without end and has no thickness. plane 3. n isometric drawing shows a corner view of a figure in 9. space 4. Through any three noncollinear points there is exactly one 9. plane Vocabulary uilder loh kus ther Word orm: The plural of locus is loci (LH cy). Related Words: location, locate efinition: locus is a set of points, all of which meet a stated condition. Main Idea: The locus of points 5 in. from a point on a plane is a circle with radius 5 in. Use Your Vocabulary Write T for true or for false. 5. square is the locus of points equidistant from a given point. T 6. sphere is the locus of points that are equidistant from a point in space. 7. circle with diameter 3 cm is the locus of points 3 cm from a given point. T 8. The locus of points 15 mi from a cell tower is a circle. hapter locus (noun)

22 Problem 1 escribing a Locus in a Plane Got It? Reasoning The sketch below shows the locus of the points in a plane 1 cm from. How would the sketch change for the locus of points in a plane 1 cm from * )? 1 cm 1 cm 9. What is the difference between * ) and? Explain. nswers may vary. Sample: * ) extends forever in opposite directions. has two endpoints. * ) 1. Would the new sketch have a parallel line segment 1 cm above * )? Yes / No 11. Would the new sketch have a parallel line segment 1 cm * below )? Yes / No 1. Would the new sketch have a parallel line 1 cm above * )? Yes / No 13. Would the new sketch have a parallel line 1 cm below? Yes / No 14. Would the new sketch have semicircles at each end? Yes / No 15. Would the new sketch have anything at each end? Yes / No 16. Sketch the locus of points in a plane 1 cm from * ). 1 cm 1 cm 17. How would the sketch change for the locus of points in a plane 1 cm from * )? nswers may vary. Sample: The new sketch would have parallel lines, not line segments. The parallel lines would be 1 cm above * ) and 1 cm below * ). The sketch would have no semicircles because the lines extend forever in opposite directions. 335 Lesson 1-6

23 Problem rawing a Locus for Two onditions Got It? What is a sketch of the locus of points in a plane that satisfy these conditions? the points equidistant from two points X and Y the points cm from the midpoint of XY Use the diagram at the right for Exercises ircle the points in a plane that are equidistant from the endpoints of a segment. circle ellipse perpendicular bisector 19. ircle the locus of points that are equidistant from X and Y. M circle with center at the midpoint perpendicular bisector X Y of XY and radius cm of XY. ircle the locus of points that are a given distance from a point. circle ellipse perpendicular bisector 1. ircle the locus of points cm from the midpoint of XY. circle with center at the midpoint of XY and radius cm perpendicular bisector of XY. Points and are the locus of points that satisfy both conditions. Problem 3 escribing a Locus in Space Got It? What is the locus of points in a plane that are equidistant from two parallel lines? omplete each statement with the appropriate word from the list below. parallel perpendicular plane 3. Points that are equidistant from two 9 lines are between the lines. 4. The locus of points in a 9 that are equidistant from two parallel lines is a line. 5. The distance between parallel lines is the length of a segment 9 to both lines. 6. What is the locus of points in a plane that are equidistant from two parallel lines? nswers may vary. Sample: The locus is a line parallel to the given lines and halfway between them. parallel plane perpendicular hapter 1 336

24 Got It? What is the locus of points in space that are equidistant from two parallel planes? 7. ross out the word phrase that does NT describe the distance between two planes. length between two points on different planes length of a segment perpendicular to each plane 8. ircle the word phrase that describes the locus of points in space that are equidistant from two parallel planes. cylinder plane halfway between the two parallel planes plane parallel to the two parallel planes Lesson heck o you UNERSTN? ompare and ontrast How are the descriptions of the locus of points for each situation alike? How are they different? in a plane, the points equidistant from points J and K in space, the points equidistant from points J and K 9. Place a in the box if the description is true for the locus of points equidistant from points J and K. Place an in the box if it is not. In a Plane escription In Space ' bisector of JK plane containing ' bisector of JK plane ' to plane containing JK 3. How are the descriptions of the loci alike? 31. How are the descriptions of the loci different? nswers may vary. Sample: oth nswers may vary. Sample: The locus contain the perpendicular bisector in a plane is a line. The locus in space of JK. is a plane. Math Success heck off the vocabulary words that you understand. locus of points plane space Rate how well you can sketch and describe a locus. Need to review Now I get it! Lesson 1-6