Chapter 14. Locus and Construction

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1 4- Constructin Parallel Lines (paes ) The distance between two parallel lines is defined as the lenth of the perpendicular from an point on one line to the other line Therefore, since CE is the perpendicular with lenth 2PQ, ever point on CD is at a distance 2PQ from AB 2 If two coplanar lines are each perpendicular to the same line, then the are parallel Thus, GH AB and GH CD The distance between two parallel lines is defined as the lenth of the perpendicular from an point on one line to the other line Thus, the distance from an point on GH to AB and CD is 2 GH 3 a Use Construction 7 b Mark an point Q on l Use Construction 6 to construct the perpendicular to m throuh Q Let R be the intersection of this perpendicular and line m Then use Construction 3 to construct the perpendicular bisector of QR c Construct a line parallel to l (and above l) that is the same distance awa as m Etend QR h to a point S such that QR RS Then use Construction 7 to construct a line parallel to l throuh S d Yes If two of three lines in the same plane are each parallel to the third line, then the are parallel to each other 4 a Use Construction to construct a sement AB conruent to the sement with lenth b Use Construction 5 to construct AC and BD perpendicular to AB with C and D on the same side of AB Set the compass radius to a With A as the center, mark off point X on AC h With B as the center and usin the same radius, mark off point Y on BD h Draw XY ABYX is a rectanle b Use Construction to construct a sement AB conruent to the sement with lenth a Use Construction 5 to construct AC and BD perpendicular to AB with C and D on the same side of AB Set the compass radius to a With A as the center, mark off point X on AC h With B as the center and usin the same Chapter 4 Locus and Construction radius, mark off point Y on BD h Draw XY ABYX is a square c Use Construction to construct a sement AB conruent to the sement with lenth a Use Construction 2 to construct an anle conruent to A on AB h n the side of this anle not containin point B, use Construction to construct a sement AC conruent to the sement with lenth b Use Construction 7 to construct a line parallel to AB throuh point C n this line, use Construction to construct CD conruent to the sement with lenth a such that CD is on the same side of AC Draw BD ABDC is a paralleloram d Repeat part c, but construct AC to be conruent to the sement with lenth a ABDC is a rhombus 5 a Construct the perpendicular bisector of AB at C Construct the perpendicular bisector of AC at D and of CB at E Then D, C, and E divide AB into four conruent parts b Set the compass radius to AB Draw a circle c Set the compass radius to AC Draw a circle 6 a Use Construction 7 b Use Construction 3 to find the midpoint M of AC Draw BMThen BM is the median to AC c Use Construction 3 to find the midpoint N of BC Draw AN Then AN is the median to BC d Yes An two medians of a trianle intersect in the same point, and two points determine a line Thus, the line drawn from C to the median of AB is the same as that drawn from C to P 7 a Use Construction 3 b Use Construction 3 c Use Construction 3 to find the midpoint M of AB d P and M An two perpendicular bisectors of the sides of a trianle intersect in the same point Thus, the line containin P and M is the perpendicular bisector of AB 8 a Use Construction 6 to construct a line perpendicular to AC throuh B intersectin AC at D BD is the altitude to AC b Use Construction 6 to construct a line perpendicular to BC throuh A intersectin BC at E AE is the altitude to BC 343

2 c P and C An two altitudes of the sides of a trianle intersect in the same point, and two points determine a line Thus, PC contains the altitude to AB 9 a Use Construction 4 to construct the anle bisector, BX h, of CBA Let D be the intersection of BX h and AC Draw BD Then BD is the anle bisector from B in ABC b Use Construction 4 to construct the anle bisector, AY h, of CAB Let E be the intersection of AY h and BC Draw AE Then AE is the anle bisector from A in ABC c P and C An two anle bisectors of a trianle intersect in the same point, and two points determine a line Thus, the line determined b P and C contains the anle bisector from C in ABC 0 b Draw an point D on AC h Set the compass radius to AD With D as the center, mark off a point X on AD h With X as the center and usin the same compass radius, mark off point E on AD h Then DE 2AD c Use Construction 7 to construct a line parallel to EB throuh D d If a line is parallel to one side of a trianle and intersects the other two sides, then the points of intersection divide the sides proportionall Therefore, since DF is parallel to EB and intersects the sides of ABE, AF : FB5AD : DE5 : 2 a Answers will var b Draw an point N on PL h Set the compass radius to PN With L as the center, mark off point L 2 on PL h With L 2 as the center and usin the same compass radius, mark off point L 3 on PL h Repeat this procedure until ou have drawn L 8 The point Q is L 8, and the point S is L 5 c Use Construction 7 to draw a line parallel to QR throuh S d If a line is parallel to one side of a trianle and intersects the other two sides, then the points of intersection divide the sides proportionall Thus, since ST is parallel to QR and intersects the sides of PQR, PS : SQ PT : TR 3 : 5 If two line sements are divided proportionall, then the ratio of the lenth of a part of one sement to the lenth of the whole is equal to the ratio of the correspondin lenths of the other sement Thus, PS : PQ PT : PR 3 : 8 Since P P, PST ~ PQR b 3 SAS~ with a constant of proportionalit of The Meanin of Locus (paes 62 63) No The locus of all points equidistant from the endpoints of a sement is the line that is the perpendicular bisector of the sement Different points on the line will be different distances from the endpoints 2 The perpendicular to the line or ra throuh the point 3 The circle with the point as the center and a radius of 0 centimeters 4 The perpendicular bisector of AB 5 The line that is parallel to both lines and midwa 6 A pair of lines parallel to AB each 4 inches awa from AB such that AB is midwa 7 The line that is parallel to both lines and midwa 8 The line that is parallel to both sides and midwa 9 The line containin the diaonal between the two other vertices 0 The point that is the intersection of both diaonals The circle with the same center and a radius of inch 2 The circle with the same center and a radius of 5 inches 3 A pair of circles, both with the same center as the iven circle and one with a radius of inch and the other with a radius of 5 inches 4 a The circle with the same center and a radius of (r m) b The circle with the same center and a radius of (r m) c A pair of circles, both with the same center as the iven circle and one with a radius of (r m) and the other with a radius of (r m) 5 The circle with the same center as the iven circles and a radius of 4 centimeters 6 The perpendicular bisector of AB (Note: The trianles are drawn on both sides of AB) 7 A pair of lines parallel to AB each 3 feet awa from AB such that AB is midwa 344

3 Applin Skills 8 The circle centered at the base of the hour hand with a radius equal to the lenth of the hour hand 9 The line parallel to the track at a distance from the track equal to the radius of the train wheel 20 The line parallel to the curbs that is equidistant from the curbs 2 The circle with the stake as the center and a radius of 6 meters 22 The line that bisects the anle formed b the two roads 23 The line that is the perpendicular bisector of the two floats on the lake 24 The line parallel to the horizontal line at a distance from the horizontal line equal to the radius of the dime 25 The circle with the same center as the circular track and a radius of 32 feet 4-3 Five Fundamental Loci (paes 65 66) Yes The locus of points equidistant to PQ and RS is the line that is parallel to both and midwa This line is the perpendicular bisector of SP, and so ever point on this line is equidistant to P and S 2 Two lines that are equidistant from two intersectin lines are the anle bisectors of the anles formed b the two intersectin lines Let one of the anles formed b the intersectin lines have measure 2 Then both of the adjacent anles have measure (80 2) Thus, the measures of the anles formed b the anle bisectors are and (90 ), and so the measure of the anle formed b the two anle bisectors is (90 ), or 90 3 The perpendicular bisector of the sement formed b the two points 4 The circle with a radius of 6 inches and the center equal to the midpoint of the sement 5 The perpendicular bisector of the base of the isosceles trianle 6 The line containin the anle bisector of the verte anle of the isosceles trianle 7 The two distinct perpendicular bisectors of the sides of the square 8 The line parallel to both bases and midwa 9 The circle with a radius of 4 inches and the center equal to the midpoint of the base 0 The locus is the union of the two sements that are both conruent to and parallel to the altitude and 6 centimeters awa from the altitude, and the two semicircles of radius 6 centimeters centered at the endpoints of the altitude a The locus is the line parallel to both lines and midwa b Draw point P on an of the iven lines The locus is the circle centered at P with a radius of 3 centimeters c 2 2 a The locus is the perpendicular bisector of AB b The locus is a circle centered at M with a radius of 2 AB c 2 d A square The diaonals bisect each other since the are both radii of the same circle Therefore, the quadrilateral formed is a paralleloram The diaonals are conruent since the are both diameters of the same circle Therefore, the paralleloram is a rhombus The diaonals are perpendicular since one is contained in the perpendicular bisector of the other Therefore, the rhombus is a square 4-4 Points at a Fied Distance in Coordinate Geometr (paes 68 69) Yes It intersects the circle in two points 2 Yes The locus of points of two intersectin lines is a pair of lines that bisect the anles formed b the intersectin lines A point is on the anle bisector of an anle if it is equidistant from the sides of the anle Since PA and PB are each tanent to the circle, the each intersect the circle in eactl one point At these two points, the radii drawn are perpendicular to the lines and meet in the point Thus, is equidistant from PA and PB, and so is on the locus 345

4 ( ) ( 2) ( 2 ) 2 ( 2 ) ( 2 3) 2 ( ) ( 3) 2 ( 2 5) and and 2 6 and and 3 ( 4, 3) and (3, 4) 4 (2, 5) and ( 5, 2) 5 ( 8, 5) and (6, 9) 6 ( 2, 0) and (0, 2) 7 a and 7 b and 3 c (, 7), (3, 7), (, ), (3, ) 8 a ( 2 2) 2 ( 2 2) b 4 and 2 c (5, 2) and (, 2) 9 a b 3 and 3 c ( 4, 3), (4, 3), ( 4, 3), (4, 3) 20 a b 8 and 8 c (8, 6), (8, 6), ( 8, 6), ( 8, 6) 2 a ( 4) ( 4) b 2 and 2 c (2, 0), (22, 4!2), (22, 24!2), ( 2, 0) 22 a ( ) 2 ( 2 5) b and 4 c (, 0), (, 0), (4, 5) 4-5 Equidistant Lines in Coordinate Geometr (paes ) Yes The locus of points equidistant from two intersectin lines is a pair of lines that are the anle bisectors of the anles formed b the intersectin lines The anle bisectors of the anles formed b and are the - and -aes 2 Yes The locus of points equidistant from two parallel lines is the line parallel to the lines and midwa The slope of this line is, and the -intercept is the averae of the -intercepts of the iven lines, that is, 2 0 b 2 6 Therefore, the locus is (, 6) and (, 0) 6 (4, ) and ( 4, ) Applin Skills 7 a b ? 2 2 (22) 5 65? c Distance from (3, ) to ( 2, 6)!50 Distance from (5, 5) to ( 2, 6)!50 8 a b ? c 5 d A(, 4) e B(5, 0) f PA5PB52!2 9 The locus of points equidistant from two intersectin lines is a pair of lines that bisect the anles formed b the intersectin lines Thus, the locus of points equidistant from and are the - and -aes Translations preserve anle measure Thus, if the - and -aes are the anle bisectors of and, then their imaes will be the anle bisectors of the imaes of and Under T 0,, the imaes of and are and, respectivel The imaes of the anle bisectors are the -ais and the line under the same translation Therefore, the -ais and the line are the anle bisectors of and, and so the are the locus of points equidistant from these two lines 20 The locus of points equidistant from two intersectin lines is a pair of lines that bisect the anles formed b the intersectin lines Thus, the locus of points equidistant from 3 and 3 are the - and -aes Translations preserve anle measure Thus, if the - and -ais are the anle bisectors of 3 and 3, then their imaes will be the anle bisectors of the imaes of 3 and 3 Under T 0,22, the imaes of 3 and 3 are 3 2 and 3 2, respectivel The imaes of the anle bisectors are the -ais and the line 2 under the same translation Therefore, the -ais and the line 2 are the anle bisectors of 3 2 and 3 2, and so the are the locus of points equidistant from these two lines 346

5 2 a (0, b) b (0, c) c M5 A 0 2 0, b 2 c B 5 A 0, b 2 c B d Since the first line is parallel to the second line, alternate interior anles are conruent Thus, /BrArM 5/BAM Since vertical anles are conruent, BMA B MA Since M is the midpoint of AAr, AM>ArM Therefore, b ASA, ABM A B M e Since correspondin parts of conruent trianles are conruent, BM>BrM The distance between two lines is the lenth of a perpendicular sement joinin both lines Since BBr is perpendicular to the iven lines, BB is the distance between the iven lines Thus, since M is midwa between B and B, it lies on the line equidistant to the iven lines 4-6 Points Equidistant from a Point and a Line (paes ) No The solutions to the equation are 2 and 4 The raph of intersects the -ais at ( 2, 0) and (4, 0) When the raph intersects the -ais, the -coordinate is 0 2 Yes The tanent of a parabola of the form 5a 2 b c is a horizontal line passin throuh its turnin point Since (, 0) is the turnin point and is on the -ais, the -ais is the tanent of the parabola 3 (3, 8); 3 4 (, 2); 5 ( 2, 5); 2 6 (, 6); 7 ( 4, 20); 4 8 A 5 2, 27 4 B ; (3, 5), (0, 2) 0 ( 2, 3), (, 0) (4, 3), (, 0) 2 ( 4, 5), (, 0) 3 (5, 7), (, ) 4 (2, 2), ( 2, 6) 5 (2, 3), (6, 5) 6 (2, 0), ( 2, 8) Hands-n Activit a 2 2 6a 9a 5 5a 2 2 2ha ah 2 k Review Eercises (paes ) a Answers will var Eample: Draw an sement and then construct its perpendicular bisector The anles formed are riht anles b Use Construction 4 to bisect the anle formed in part a c Use Construction to construct AB conruent to the sement with lenth a Use Construction 2 to construct ABX conruent to the 347

6 45 anle from part b Set the compass radius to b With B as the center, mark off a point C alon BX h Use Construction 7 to construct a line parallel to BC throuh A and a line parallel to AB throuh C Let D be the intersection of these two lines Then ABCD is a paralleloram with m B 45 2 Draw a ra PX h and a sement YZ Use Construction to construct a sement conruent to YZ on PX h Call the other endpoint Q Repeat this construction on,,, and Q 4 X Q Q 2 X X Q 3 X Let the final endpoint be called Q Rename Q 2 as S Then PS : SQ 2 : 3 3 The perpendicular bisector of the sement formed b the two points 4 The two points that are on the perpendicular bisector of AB and!7 centimeters from the sement 5 The two points that are on the perpendicular bisector of the sement and 2 centimeters from the sement 6 The two points that are the intersection of the line parallel to the two lines and 25 inches from each of them and the circle centered at the iven point with radius 4 inches 5 (4, 3) and ( 3, 4) 6 (2, 3) and (2, ) 7 ( 2, ) and (, 2) 8 (4, 2) and (, 5) 7 ( ) 2 ( 2 2) (4, 4) and (0, 4) ais and -ais (4, 3) and (3, 4) 4 a b c (3, 4) and (, 4) 9 point 20 (3, ) Eploration (pae 632) a An ellipse is the locus of points such that the sum of the distances from two fied points is a constant, k Let k represent the lenth of the strin Let P be an point on the curve that is drawn Then, F P PF 2 5 k, the lenth of the strin Since P is an arbitrar point on the curve, all of the points on the curve are such that the sum of the distances from F and F 2 is k Therefore, the curve drawn is an ellipse 348

7 b The ellipse becomes more circular c The distance between each point and the curve can increase infinitel while keepin the difference between the two distances constant Cumulative Review (paes ) Part I Part II In ABC, A A b the refleive propert of conruence If two parallel lines are cut b a transversal, then the correspondin anles are conruent Since DEBC, ADE ABC Therefore, ABC ADE and the sides are in proportion AD AB 5 AE AC AE 20 AE58 EC AC AE The circumference of the base 75 Then the radius is: 2pr575 r5 2p 75 Part III 3 a b Volume5 3 pr2 h c (3, 8) and (, 4) ( 2 3)( ) , pA 2pB pA 5,625 4p B 2 5 5,625 p <,800 cubic feet 4 a (4 2 ) ( 2 2) The line intersects the circle onl at (2, 2) so is tanent to the circle b The slope of the tanent line is The center of the circle is (0, 0), so the slope of the radius 2 to the point of tanenc is 2 5 Since these slopes are neative reciprocals, the tanent and the radius to the point of tanenc are perpendicular Part IV 5 a Radii of conruent circles are conruent If two points are each equidistant from the endpoints of a line sements, then the points determine the perpendicular bisector of the line sement AE>AD>BE>BD AM>MB DM>ME AE>AD>BE>BD /EAM>/DAM>/EBM>/DBM /AEM>/BEM>/ADM>/BDM /AEB>/ADB /EAD>/EBD /AME>/EMB>/AMD>/BMD b Radii of conruent circles are conruent ABC DEF b SSS Correspondin parts of conruent trianles are conruent AC>DF AB>DE BC>EF /FDE>/CAB /ACB>/DFE /ABC>/DEF 349

1 What is the solution of the system of equations graphed below? y = 2x + 1

1 What is the solution of the system of equations graphed below? y = 2x + 1 3 As shown in the diagram below, when hexagon ABCDEF is reflected over line m, the image is hexagon A'B'C'D'E'F'. y = x 2 + 2x