Coordinate Graphing and Geometric Constructions

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1 HPTER 9 oordinate Graphing and Geometric onstructions hapter Vocabular coordinate plane origin graph image line of reflection rotation midpoint -ais ordered pair quadrants translation line smmetr rotational smmetr perpendicular bisector -ais coordinates transformation reflection point smmetr construction altitude 9. Using oordinates to Graph Points Two perpendicular number lines can be used to form a sstem for locating points called a coordinate plane. The horizontal line is called the -ais. The vertical line is called the -ais. The point where the aes cross is called the origin, and this point is represented b the ordered pair (0, 0). 7

2 Quadrant II p(, ) - ais Quadrant I origin Quadrant III Quadrant IV -ais Other points are represented b ordered pairs according to their distance with respect to each ais. The ordered pair (, ) corresponds to on the -ais and on the -ais. Each ordered pair is a set of coordinates for the point it names. point is the graph of its ordered pair. Start at the origin (0, 0). To graph an ordered pair Read the first number (the -coordinate) and move left or right the number of units indicated. Read the second number (the -coordinate) and move up or down the number of units indicated. Place a dot at the location and label the point with a capital letter. The -ais and the -ais divide the graph into four regions called quadrants. The signs of the coordinates in each quadrant are: Quadrant I: (, ) Quadrant II: (, ) Quadrant III: (, ) Quadrant IV: (, ) 7 hapter 9 oordinate Graphing and Geometric onstructions

3 Model Problems. Use an ordered pair to name the location of each point. a. P b. Q c. R d. S R S Solution a. units right and units up P(, ) b. units left and units up Q(, ) c. units left and units down R(, ) d. units right and units down S(, ). Name the point that has the given coordinates. Give the quadrant for each point. H Q G J F E I D P Using oordinates to Graph Points 7

4 Solution oordinates Point Quadrant a. (, 0) F none; -ais b. (, ) IV c. (, ) H II d. (0, ) J none; -ais e. (, ) E III f. (, ) I. Graph each pair of points and connect them with a line segment. Identif the relationship of the segments. a. P(, ) and Q(, ) R(, ) and S(, ) b. L(, ) and M(, ) N(, ) and O(, ) Solution a. Points P and Q, which have the same -coordinate, are on a line that is parallel to the -ais. Points R and S, which have the same - coordinate, are on a line parallel to the -ais. Lines parallel to the -ais are parallel to each other. So, PQ RS. Q R S b. Points L and M, which have the same -coordinate, are on a line parallel to the -ais. Lines that are parallel to the -ais are perpendicular to the -ais and to lines parallel to the -ais. So, LM NO. M N L P O 7 hapter 9 oordinate Graphing and Geometric onstructions

5 Practice Multiple-hoice Questions. Which ordered pair locates a point on the -ais?. (, ). (0, ). (, ) D. (6, 0). Which ordered pair locates a point on the -ais?. (0, ). ( 6, ). (0, 0) D. (8, 0). Points K(, 7) and L( 6, 7) lie on a line that. is parallel to the -ais. passes through the origin. is parallel to the -ais D. passes through Quadrants III and IV. The ordered pair for the point that is units left and units up from point P is P 6 7. (, ). (, 7). (, ) D. (, ) 7 6. Which set of points is on a line perpendicular to the -ais?. P(, 6), Q(, ), R(0, 0). S(6, 9), T(6, ), U(6, ). V(, 8), W( 7, 8), X(0, 8) D. K(0, ), L(, ),M(8, ) 6. Point Q is units right and units down from which point? Q. W(, ). X(, ). Y(, ) D. Z(, ) 7. Which point does NOT lie on either the -ais or the -ais?. L(0, ). M( 7, 0). N(, ) D. O(0, 0) 8. Which ordered pair comes net in the pattern? (, ), (, ), (, ), (, ), (?,?). (6, 6). (6, 7). (7, 6) D. (7, 8) 7 6 Using oordinates to Graph Points 7

6 Short-nswer Questions 9. Draw and label a pair of coordinate aes. Graph the point that corresponds to each ordered pair. Label each point with its coordinates. a. (, ) b. (8, 0) c. ( 6, ) d. (0, ) e. (, 7) f. (.,.) 0. Name the quadrant for each ordered pair. a. ( 9, 6) b. ( 7, ) c. (, ) d. (0, ) e. (, ) Open-Response Questions. Graph these points and connect them with a line segment: P(, ) and Q(, ). a. Give the coordinates of two points R and S that are on a line perpendicular to PQ. Graph RS. b. Give the coordinates of two points V and W that are on a line parallel to RS. Graph VW.. Write the letter and coordinates of the points graphed that meet each condition given. 6 H I 7 6 E P N G M 6 7 F K J L 6 7 a. The -coordinate is greater than the -coordinate. b. The -coordinate is the opposite of the -coordinate. c. The -coordinate and the -coordinate are equal.. a. P(, ) is in Quadrant II. In which quadrant is Q(, )? b. S(, ) is in Quadrant I. In which quadrant is T(, )? c. M(, ) is in Quadrant III. In which quadrant is N(, )?. a. Which point does NOT fit the pattern? Eplain wh the point does not fit and change the coordinates so that it does. (, ), (, ), (, 7), D(, ), E(, ), F(6, 6) b. Give the ordered pairs for two more points that fit the pattern. D 9. Distance etween Two Points on a oordinate Plane The distance between two points is the length of the line segment that has these points as endpoints. The distance between two points is alwas a positive number. 76 hapter 9 oordinate Graphing and Geometric onstructions

7 Line segments on the coordinate plane are horizontal, vertical, or diagonal. The length of a horizontal line segment ƒ difference of -coordinates ƒ. The length of a vertical line segment ƒ difference of -coordinates ƒ. To find the length of a diagonal line segment, form a right triangle with the diagonal as its hpotenuse. Then, find the coordinates of the verte of the right angle. The legs are horizontal and vertical line segments, so their lengths can be found using the rules above. Use the Pthagorean Theorem to find the length of the diagonal line segment that corresponds to the hpotenuse of the right triangle (, ) (, 6) (, ) same - coordinate same -coordinate Model Problems. Find the distance between each pair of points a. P(, ) and Q(, ) b. R(6, ) and S(6, ) 7 6 P(, ) Q(, ) R(6, ) S(6, ) 6 7 Distance etween Two Points on a oordinate Plane 77

8 Solution a. The line segment is horizontal, so: length ƒ difference of -coordinates ƒ ƒ ( ) ƒ ƒ ƒ ƒ 7 ƒ 7 units b. The line segment is vertical, so: length ƒ difference of -coordinates ƒ ƒ ƒ ƒ 6 ƒ 6 units You can verif both lengths b counting units on the graph.. Find the distance between (, ) and (, ). 6 (, ) (, ) (, ) Solution The line segment is diagonal, so ou must form right triangle with (, ) and use the Pthagorean Theorem. length of vertical leg ƒ difference of -coordinates ƒ ƒ ƒ ƒ ƒ length of horizontal leg ƒ difference of -coordinates ƒ ƒ ( ) ƒ ƒ ƒ ƒ ƒ Use the Pthagorean Theorem: c 6 9 c c nswer The length of is units.. Graph the following points: D(, ), E(, 6), and F(, ). Then draw DEF and find its area. Solution 7 6 E (, 6) F (, ) D (, ) G (, ) hapter 9 oordinate Graphing and Geometric onstructions

9 To find the area, calculate the length of the base DF. Draw EG perpendicular to DF and find the height of the triangle. DF ƒ ( ) ƒ ƒ ƒ ƒ 6 ƒ 6 EG ƒ 6 ƒ ƒ ƒ Substitute in DF and EG for the base and bh (DF)(EF) height. 6 nswer The area of DEF is square units. Practice Multiple-hoice Questions. The distance between which pair of points is 7 units?. (0, 7) and (7, 0). (, ) and (, ). (, ) and (, ) D. (, 7) and (, 7). What is the distance between (, ) and (7, )?. units. units. units D. 9 units. What is the area of the figure formed when the points J(, ), K(, 6), L(, 6), and M(, ) are graphed and connected in order?. 6 units. 0 units. units D. units. Diego graphed these points: Q(, ), R(, ), and S(, ). Which point must be graphed to complete a rectangle?. T(, ). T(, ). T(, ) D. T(, ). What kind of polgon is formed when these points are graphed and connected in order? W(0, ), X(, ), Y(6, ), Z(, ). rectangle. rhombus. parallelogram D. trapezoid 6. What is the area of the polgon formed when these points are graphed and connected in order? J( 8, 7), K(9, 7), L(9, ). 0 square units. 6 square units. 0 square units D. 8 square units Distance etween Two Points on a oordinate Plane 79

10 7. What is the length of the diagonal of the figure formed when these points are graphed and connected in order? P(, ), Q(, ), R(, ), S(, ). 7 units. 0 units. units D. units 8. The distance from (9, ) to the origin is. " units. units. "6 units D. "06 units Short-nswer Questions 9. Find the distance between each pair of points. a. (, ) and (, ) b. ( 6, ) and ( 6, 6) c. (, ) and (0, 8) 0. The distance between (, 7) and (, ) is units. What is the value of?. The distance between (, ) and (, ) is 7 units. What is the value of? Open-Response Questions. a. Graph points (, ), (, ), (, ), and D(0, ). b. Identif the tpe of quadrilateral. c. Find the area of D.. a. Give the coordinates for a set of points that form a square with an area of 00 square units. One of the points must be W(, ). b. Find the length of the diagonal of the square to the nearest tenth.. a. Graph these points: P(6, 0), Q(, 6), R(, 6), S( 6, 0), T(, 6), and U(, 6). b. Identif the polgon formed. c. Find the lengths of QR and QP. Round to the nearest tenth if necessar. d. Is the polgon regular? Eplain. e. Eplain how ou could find the area of PQRSTU. arr out our plan, showing all steps. 9. Translations transformation is a wa of moving a geometric figure without changing its size or shape. The figure that results after the move is called the image of the original figure. For each point of the original figure, there is a corresponding point of the image. Imagine sliding a chair across the floor so that each leg moves the same distance in the same direction. This is an eample of a translation. 80 hapter 9 oordinate Graphing and Geometric onstructions

11 translation (or slide) moves ever point of a figure the same distance in the same direction. Triangle is the translation image of triangle. and the two triangles are congruent and have the same orientation. In the figure below, PQRS is translated b moving ever point units to the right and units down. P Q R S units units Q 9 0 R P S To find the corresponding vertices of the image: add to each -coordinate. add to each -coordinate. P(, ) S P (6, ) Q(, ) S Q (8, ) R(, ) S R (9, ) S(, ) S S (9, ) Finding the coordinates of a translation image Under a translation of a units in the horizontal direction and b units in the vertical direction, the image of P(, ) is P ( a, b). Translations 8

12 Model Problems. Graph the image of XYZ with vertices X(, ), Y(, 0), and Z(, ) after a translation 6 units left and units up. X 6 Z X Y Y Z Solution dd 6 to the -coordinate of each verte. dd to the - coordinate of each verte. X(, ) S X ( 6, ) S X (, 6) Y(, 0) S Y ( 6, 0 ) S Y (, ) Z(, ) S Z ( 6, ) S Z (, ). The coordinates of D are (, ), (, ), (, ), and D(, ). fter a translation, the image of is (6, ). Find the coordinates of,, and D after this same translation and graph D D D 8 hapter 9 oordinate Graphing and Geometric onstructions

13 Solution Since (, ) S (6, ), ou can find the numbers that were added to each coordinate. a 6, so a b, so b 7 dd to each -coordinate and 7 to each -coordinate of the other vertices. (, ) S (, ( 7)) S (8, ) (, ) S (, ( 7)) S (8, 9) D(, ) S D (, ( 7)) S D (6, 8) Practice Multiple-hoice Questions. F GH is a translation of FGH F H G F H. right units, up units. left units, down units. left units, down units D. right units, up units. The coordinates of WXYZ are W( 6, ), X(, ), Y(, ), and Z(, ). fter a translation 8 units right and units down, the coordinates of the image are. W (, ), X (, 0), Y ( 0, 0), Z (, ). W (, ), X (, 0), Y (6, 0), Z (, ). W (, ), X (, 6), Y (6, 6), Z (, ) D. W (, ), X (, ), Y (6, ), Z (, ) G. There are twelve congruent plates. fter a translation, the image of plate is plate 8. fter the same translation, what is the image of plate?. plate 9. plate 0. plate D. plate. Which pair of figures shows a translation?... D Translations 8

14 . Which of the numbered figures are translations of the shaded figure?. onl.,,, and 8 onl., 6, 8, and 0 onl D.,, 8, and 0 onl Short-nswer Questions For 6 and 7, cop each figure onto graph paper. Then graph the image of each figure after a translation 6 units to the right and units down the same set of aes after each translation. a. units right and units down b. units left and units up c. units right and 6 units down 9. is the image of after a translation 7 units left and units up. Graph before the translation op the figure shown onto graph paper. Graph the image of the figure on Open-Response Questions 0. The coordinates of MNOP are M(0, 6), N(, 6), O(7, ), and P(, ). a. Give the coordinates of the image after a translation units left and units up. b. Graph MNOP and M NOP. 8 hapter 9 oordinate Graphing and Geometric onstructions

15 . The coordinates of are (, ), (6, 6), and (7, ). fter a translation, the image of verte is ( 6, ). a. Give the coordinates of and after the same translation. b. Graph and.. The coordinates of DEFG are D(, ), E(, ), F(, ), and G(, ). a. Describe a translation that will move verte E to the origin. b. Give the coordinates of D, E, F, and G after the translation described in part a. c. Graph DEFG and D EFG. 9. Reflections and Smmetr reflection is a transformation in which a figure is flipped or reflected over a line of reflection to produce a mirror image. The figure and its image are congruent, but have opposite orientations. Each point and its image are the same distance from the line of reflection. fter reflection in the -ais, the image of P(, ) is P (, ). is the reflection of in the -ais. Each -coordinate of the image is multiplied b. (, ) S (, ) (, ) S (, ) (, ) S (, ) fter a reflection in the -ais, the image of P(, ) is P (, ). is the reflection of in the -ais. Each -coordinate of the image is multiplied b. (, ) S (, ) (, ) S (, ) (, ) S (, ) Reflections and Smmetr 8

16 figure has line smmetr if it is possible to draw a line that cuts the figure into two parts such that one part is a mirror image of the other. figure ma have one, none, or several lines of smmetr. GH and F are lines of smmetr for heagon DEF. G F E H D figure has point smmetr if for ever point in the figure, there is another point at the same distance from the center on the opposite side. The center is the midpoint of the line segment joining the pair of points. enter of smmetr Figures ma have onl line smmetr, onl point smmetr, or both, or no smmetr at all. Model Problems. Find the image of DEF with vertices D(, ), E(, ), and F(6, ) after a reflection: a. in the -ais b. in the -ais D F 6 E 86 hapter 9 oordinate Graphing and Geometric onstructions

17 Solution a. For a reflection in the -ais, each -coordinate is multiplied b. D(, ) S D (, ) E(, ) S E (, ) F(6, ) S F ( 6, ) 6 F E 6 D D F 6 E b. For a reflection in the -ais, each -coordinate is multiplied b. D(, ) S D (, ) E(, ) S E (, ) F(6, ) S F (6, ) 7 6 D 6 E D F 6 E F. Sketch the image of after a reflection in point. Solution Use a ruler to measure the distance from to. Since is the center of reflection, the image of is on the same distance as from, but on the opposite side. Label the point. Reflections and Smmetr 87

18 Repeat the measuring to locate, placing the ruler along. Draw the image of.. Determine what kind of smmetr KLMN has. K L N M Solution The parallelogram does not have line smmetr. None of the lines drawn allow the parallelogram to be folded so that points on one side of the line will coincide with points on the other side. K L No lines of smmetr N M K L P Point smmetr N M nswer KLMN has point smmetr with P, the intersection of its diagonals, as the center of smmetr. 88 hapter 9 oordinate Graphing and Geometric onstructions

19 Practice Multiple-hoice Questions. How man lines of smmetr does the figure have?. 0.. D.. Which figure does NOT have point smmetr?... D.. Describe how XYZ was transformed to produce X YZ. X Z Y. reflection in -ais. translation 8 units down. reflection in -ais D. translation units down X Y Z. Describe how KLMN was transformed to produce K LMN. M N L K K. reflection in the -ais. translation left units. reflection in the -ais, then translation right units D. translation right units, then reflection in the -ais. The coordinates of the vertices of PQR are P(, ), Q(, ), and R(, ). fter a reflection in the -ais, the coordinates of the vertices are. P (, ), Q (, ), R (, ). P (0, ), Q (0, ), R (0, ). P (, ), Q (, ), R (, ) D. P (, ), Q (, ), R (, ) 6. The coordinates of the endpoints of a line segment are S(, 7) and T(, 8). fter a reflection in the -ais, followed b a reflection in the -ais, the coordinates of the image are. S (, 7), T (, 8). S (, 7), T (, 8). S (, 7), T (, 8) D. S (, 7), T (, 8) M N L Reflections and Smmetr 89

20 7. Which pair of transformations results in the same image as a translation down units followed b a reflection in the -ais?. reflection in the -ais, followed b a translation units up. reflection in the -ais, followed b a translation units down. translation up units, followed b a reflection in the -ais D. reflection in the -ais, followed b a translation units left 8. Which figure has both line smmetr and point smmetr?... D.. How man lines of smmetr does each figure have? a. square b. regular pentagon c. regular heagon d. regular polgon of n sides. Sketch the image of JKLM after a reflection in point L. M J Open-Response Questions. a. Graph the triangle with vertices D(, ), E(, ), and F(, ). b. Reflect DEF in the -ais. Give the coordinates of D, E, and F. c. Reflect D EF in the -ais. Give the coordinates of D, E, and F.. manda said that all regular polgons have both line smmetr and point smmetr. State whether manda is or is not correct. Make drawings to support our conclusion. L K Short-nswer Questions 9. op the figure. Then show how to complete the figure so that MN is a line of smmetr. M 0. Draw a triangle with the given number of lines of smmetr. a. 0 b. c. N 90 hapter 9 oordinate Graphing and Geometric onstructions

21 . a. Graph the quadrilateral (0, ), (, ), (0, ), D(, ). b. Reflect D in the -ais. Describe what ou see. Write the coordinates of the vertices of D. c. Give a reason for what ou observed in part b. 6. a. Draw equilateral triangle. b. Draw l, a line of reflection for which the image of is. c. Draw m, a line of reflection for which the image of is. 9. Rotations rotation is a transformation that turns a figure about a point. When rotating a figure, ou need: a center of rotation about which to rotate the figure. a clockwise or counterclockwise direction of rotation. a number of degrees of rotation. 90º T Figure is the rotation image of figure. Point T is the center of rotation. Figure was rotated clockwise 90. Trace figure and, without moving the paper, put our pencil point on point T. Turn the paper until figure matches figure. When a figure can be rotated a certain number of degrees about a center point so that the image fits perfectl on top of the original figure, the figure has rotational smmetr. n regular plogon has rotational smmetr. F 60º E D When regular heagon DEF is rotated 60 or 60 about its 6 center, the image appears to be in eactl the same position as the original figure. Verte has rotated to position, to, to D, and so on. The heagon fits over its original position 6 times in the process of a complete rotation (60 ). Rotations 9

22 Model Problems. Graph the quadrilateral with vertices (, ), (, ), (, ), and D(, ) and its image after a 90 counterclockwise turn about the origin. Solution 6 D D 90º (, ) S (, ) (, ) S (, ) (, ) S (, ) D(, ) S D (, ). Give the measure of the smallest angle each figure can be rotated to fit over its original position. Mark the center of rotation. a. b. Solution a. The parallelogram would fit over itself after a 80 rotation either clockwise or counterclockwise. b. The figure would fit over itself after a rotation of 60 clockwise or 0 counterclockwise. a. T b. T 9 hapter 9 oordinate Graphing and Geometric onstructions

23 Practice Multiple-hoice Questions. Figure Y is the image of figure X. Identif the transformation. Y X. Which figure would require a complete turn of 60 to fit over itself?... D.. translation left. reflection in the -ais. rotation clockwise 90 D. rotation counterclockwise 90. rotation counterclockwise of 90 is equivalent to. reflection in the -ais. rotation clockwise 70. reflection in the -ais D. translation left and down the same number of units. Which is the image of the figure shown after a 90 clockwise rotation about point P?... D. P. For the figure shown, what is the measure of the smallest angle of rotation about T that would allow the image to fit over itself? D Which of the following pairs of transformations alwas brings a figure back to its original position?. reflection in the -ais, then reflection in the -ais. rotation 90 clockwise, translation up. reflection in the -ais, then reflection in the -ais D. reflection in the -ais, rotation 90 clockwise 7. M N is the rotation image of MN about T. What is the angle of rotation in a clockwise direction? M T N T M N D. 00 Rotations 9

24 8. The area of KLM is square units. fter a 70 counterclockwise rotation about the origin, the area of image K LM is. 8 square units. 6 square units. square units D. square units Short-nswer Questions 9. Graph the image of the figure shown after a rotation 90 clockwise about (, ) Open-Response Questions. a. Graph the quadrilateral with vertices (, ), (, ),(, ), and D(, ). b. Graph the image of D after a 90 rotation clockwise about the origin. c. Graph the image of D after a 90 rotation counterclockwise about (, ).. a. Graph the triangle with vertices R(, ), S(, ), and T(, ). b. Graph the image of RST after a 80 clockwise rotation about the origin. Give the coordinates of R, S, and T. c. Describe another wa RST could have been transformed to produce the same image R ST.. What is the image of each of the given points after a rotation 80 clockwise about T? D K 0. Figure DEFG has been rotated about T. Identif the angle of rotation if the turn was a. clockwise b. counterclockwise E G F T L I H J D E D G F G T E F a. b. D c. F d. K. a. Draw a regular pentagon. b. Mark the point about which the figure can be rotated to fit over itself. c. Find the measure of the smallest angle of rotation that will allow the figure to fit over itself. 9 hapter 9 oordinate Graphing and Geometric onstructions

25 9.6 onstructing and isecting ngles construction is a drawing of a geometric figure that is made using onl a compass and an unmarked straightedge. Two of the most fundamental constructions are constructing an angle congruent to a given angle and bisecting an angle (dividing the angle into two congruent parts). Eample onstruct an angle congruent to given angle. G J D F E H D F E Step Step > Step Use a straightedge to draw an ra. Place the compass tip on D and draw an arc intersecting DE > DE. Label the intersection F. Step Using the same compass width, place the tip on, and draw an arc intersecting both ras. Label the intersections G and H. Place the tip on H. djust the compass to draw an arc through G. Step Using the same compass width, place the tip on F, and draw a second arc intersecting the first. Label the intersection J. Draw DJ. ngle JDE is congruent to angle. jjde j. Step Eample isect jxyz. Y X Z Y P Q X Z Y P Q X Z Y P Q X R Z Step Step Step Step With the compass tip on Y, draw an arc intersecting both ras. Label the intersections P and Q. Step With the compass tip on P, draw a second arc inside the angle as shown. (djust the compass width if necessar to do this.) Step Using the same compass width, place the tip > on Q and draw a third arc as shown. Label the intersection R. Draw. Ra YR > YR bisects jxyz. jxyr jryz. onstructing and isecting ngles 9

26 Practice Short-nswer Questions. For, trace each angle, then construct a congruent angle using a compass and straightedge. D. E F 6. G. D I H E F For 7 9, use a protractor to draw an angle with the given measure. isect the angle using a compass and straightedge G H I Draw a square. isect each angle of the square and etend the bisectors. Describe our observations. For 6, trace each angle, then bisect it using a compass and straightedge.. Open-Response Questions. a. Draw an obtuse angle. Then use a compass and straightedge to divide the angle into four congruent parts. b. isect one of the angles that resulted from our work in part a. What is the relationship between the measure of one of the resulting angles and the measure of the original obtuse angle? 96 hapter 9 oordinate Graphing and Geometric onstructions

27 . Draw a parallelogram that is NOT a rectangle. isect each angle of the parallelogram and etend the bisectors. Write a description of our observations. 9.7 onstructing Perpendicular Lines The midpoint of a line segment is the point that separates it into two congruent line segments. The perpendicular bisector of a line segment is a line, ra, or line segment that is perpendicular to a line segment at its midpoint. n altitude of a triangle is a line segment from a verte of the triangle perpendicular to the opposite side or to a line containing that side. Eample onstruct the perpendicular bisector of. E D D Step Step Step Step Open the compass a little more than half the length of. With the compass tip on, draw an arc intersecting. Step Use the same compass width. With the tip on, draw an arc intersecting. Label < > the points of intersection and D. Step Draw using a straightedge. Label point E. < > D bisects. and E is the midpoint of. < D E E > < > is perpendicular to.. < D > D D is the perpendicular bisector of. onstructing Perpendicular Lines 97

28 < RS > < Eample > onstruct a perpendicular to from point P not on RS. P P P P R L M S R L M S R L M S N R L M S N Step Step Step Step < > Step With the tip of the compass at P, draw an arc that intersects RS at L and M. Step With the compass tip at L, open the < compass > a little more than half the length of LM. Draw an arc below RS. Step With the compass < tip > at M and the same width used in Step, draw a second arc below RS that intersects the arc drawn in Step. Label the point of < intersection > < > N. < > < > < > Step Draw PN. PN is perpendicular to RS. PN RS. Practice Short-nswer Questions For, trace each line segment, then construct its perpendicular bisector.. For 7, trace each figure, then construct a line from point P that is perpendicular to the given line.. P. G H 6. P D. E F I. Use a ruler to draw a line segment of length 7 cm. onstruct its perpendicular bisector using a compass and straightedge. 7. P J K L 98 hapter 9 oordinate Graphing and Geometric onstructions

29 < > 8. Draw a line, XY, and a point K not on the < line. > onstruct a line perpendicular to XY through K. For 9 and 0, use the steps of Eample to construct an altitude for each triangle from point to. 9.. Trace each triangle and carr out the following steps for each. I. III. II. 0. Open-Response Questions. Eplain how ou could use the methods of this section to construct a pair of parallel lines. omplete the construction, showing each step. a. isect each side of the triangle. The bisectors will meet at a point. Label this point P. b. Measure the distance from point P to each verte of the triangle corresponding to P. For each triangle, what do ou observe about the measurements? c. How is the position of the intersection point of the perpendicular bisectors of the sides related to the tpe of triangle? hapter 9 Review Multiple-hoice Questions. The points P(0, ), Q(0, ), R(, ), and S(, ) are graphed on a coordinate plane and the points are connected in order. What is the area of PQRS?. square units. 0 square units. square units D. 0 square units. Which figure is net in the pattern?... D. Review 99

30 . The figure shown is reflected in the - ais. The image is. Which figure has eactl lines of smmetr?... D. X 6. Which figure does NOT have point smmetr?..... D. X X 7. For which of the points graphed is the -coordinate less than the -coordinate?. D.. What is the distance between points M and N? M. 8 units. 0 units. units D. 6 units X N X S W. S, V, and Z onl. S, W, and X onl. T, U, and Y onl D. V, W, and Z onl 8. When the -coordinate is positive and the -coordinate is negative, the ordered pair locates a point in which quadrant?. I. II. III D. IV V Z X Y U T 00 hapter 9 oordinate Graphing and Geometric onstructions

31 9. point in Quadrant I is reflected in the origin. The image of the point is in which quadrant?. I. II. III D. IV 0. PQ is translated so that Q is at the origin. The coordinates of P are. (0, ). (, ). ( 8, ) D. (, ) Short-nswer Questions. Trace. Use a straightedge and compass to construct the perpendicular bisector.. Trace angle DEF. Use a straightedge and compass to bisect it. F Q P Open-Response Questions. Graph the image of figure S after each transformation. S a. translation right units and down units b. reflection in the -ais c. rotation 80 clockwise about the origin. square has a diagonal with endpoints G(, ) and E(, ). a. Graph the square and give the coordinates of its other vertices, D and F. b. Find the perimeter of the square. 6. n ant was crawling on the lines of a coordinate plane. The ant started at (, 6) and went to (, 6), turned and went to (, ), then to (, ), then to (, 0), then to (, 0), then to (, ), then to (0, ), then to (0, ), and finall to (, ). a. Graph the ant s path. b. What was the total length of the ant s journe? E D. Line l contains the points L(, ) and M(, ). Line t contains the points S(, ) and T(, ). a. Graph the lines. b. How are the lines related? Review 0

32 7. a. Graph the quadrilateral with vertices (, ), (, ), (7, ), and D(8, ). b. Graph the image of D after reflection in the line containing. Give the coordinates of each verte of D. 8. Tell if the capital letter shown has: a. line smmetr. If so, draw all lines of smmetr. b. point smmetr. If so, identif the center of smmetr. c. rotational smmetr. If so, identif the measure of the smallest angle that will rotate the figure to fit over itself. 9. Find the coordinates of two points that are the same distance from the origin, but NOT on either ais. Give the distance. Show all work. 0. Eamine the dot pattern for the first four triangular numbers. st nd rd th a. Draw the fifth triangular number on a coordinate plane. Place one verte point at (0, 0). Give the coordinates of the other two verte points. b. How man points make up the fifth number? hapter 9 umulative Review Multiple-hoice Questions. What is the surface area of a cube that has side lengths of. cm?. 6.6 cm cm.. cm D. 9.0 cm. What is the volume of the figure shown?. Which is equivalent to 0,000 cm?. km. 000 cm. 0. km D. 0 m. Which measure is the greatest?. gal. 7 c. 9 qt D. 600 fl oz. What does this construction show? 6 cm cm 8 cm. 8 cm. 60 cm. 6 cm D. 70 cm. congruent segments. perpendicular bisector. bisected angle D. parallel lines 0 hapter 9 oordinate Graphing and Geometric onstructions

33 6. What is 7% of 80?... 0 D YES is similar to HOW. The perimeter of HOW is Y. $.6. $.. $.8 D. $.9 0. Find the value of the epression when 8 and 0. 6 ( ) D. 0 8 cm cm Short-nswer Questions. Find all the whole-number factors less than for the number 6,90,. E H cm S. The graph shown represents Ja s walk from his home to his friend s house mile awa. Write a brief description of Ja s walk that would match the graph. Distance O 9 cm W. cm.. cm. 6 cm D. 8. cm 8. When a number is divided b 0, the result is. Find the number.... D Use the price list shown. How much change should ou get from $0 if ou bu pens, ruler, and folders? Sam s Supplies (prices include ta) Pens for $.9 Rulers for $.88 Folders for $.66 Notebooks for $.98 Time. Jane needs to reduce the size of a photo that is inches tall and inches wide to fit in an advertisement space for the school newspaper. If the photo s height is reduced to inches, what will the width be so that the photo is proportional to the original size? Open-Response Questions. tractor wheel, including the tire, has a radius of length 7. inches. a. Find the circumference of the wheel. Use. for. b. How man times does the wheel go around if the tractor travels mile? Round our answer to the nearest whole number. Show our work. umulative Review 0

34 . The bar graph shows the number of middle school students living in four different towns. Students 7,000 6,000,000,000,000,000 Number of Middle School Students (00) c. Name the irrational numbers. d. Give a rational number that would be graphed between G and H. 8. a. How man equilateral triangles of all sizes are there in this equilateral triangle of length? b. How man would there be in an equilateral triangle of length?,000 0 rosb Salton Nevins Town Yardle a. pproimatel what percent of the total number of students live in Salton? b. If the number of middle school students living in Yardle is predicted to increase b % in 00, about how man students will be epected to live there? c. If rosb epects to have onl about,800 students in 00, what is the percent change from 00 to 00? Round to the nearest tenth. 6. Five vehicles are in line at a tollbooth. The minivan is paing its toll. The motorccle is two places behind the truck. The bus is ahead of the sports car, which is fifth in line. Which vehicle will pa its toll net? 7. Points through J on the number line represent the following numbers:.,.8,,, 0, ", 7,,, "9 Point E represents 0 and point G represents. G D E F H I J a. Write the number to which each point corresponds. b. Name the rational numbers. 9. Mr. Wu begins work at 7:0.M. and leaves at : P.M. He works Monda through Frida and takes an unpaid lunch break of hour each da. He is paid for holidas and sick das. His hourl wage is $.0. How much does Mr. Wu earn on a earl basis? Show our work. 0. a. Graph the reflection of in the origin. b. Give the coordinates of the vertices of the image hapter 9 oordinate Graphing and Geometric onstructions

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