The Coordinate Plane. #a 2 1 b 2

Size: px

Start display at page:

Download "The Coordinate Plane. #a 2 1 b 2"

Timothy Wheeler
6 years ago
Views:

1 -8 he oordinate lane LGE -8. lan What You ll Learn o find the distance between two points in the coordinate plane o find the coordinates of the midpoint of a segment in the coordinate plane... nd Wh o find the distance between two points on a map, as in Eample heck kills You ll Need for Help kills Handbook 9 pages 75 and 75 Find the square root of each number. ound to the nearest tenth if necessar lgebra Evaluate each epression for m and n 7.. (m - n) (n - m) 00. m +n 58 lgebra Evaluate each epression for a and b (a - b) a b #a b Objectives o find the distance between two points in the coordinate plane o find the coordinates of the midpoint of a segment in the coordinate plane Eamples Finding Distance eal-world onnection Finding the Midpoint Finding an Endpoint Math ackground Finding Distance on the oordinate lane Vocabular ip o help review terms shown here, see oordinate plane in the Glossar. You can think of a point as a dot, and a line as a series of points. In coordinate geometr ou describe a point b an ordered pair (, ), called the coordinates of the point. he coordinates of point are. Quadrant II O(0, 0) Quadrant III -ais Quadrant I Origin -ais Quadrant IV he coordinates of point are (5, ). You can use the uler ostulate to find the distance between two points if the points are on a horizontal line or a vertical line.o find the distance between two points that are not on a horizontal or vertical line, ou can use the Distance Formula. Ke oncepts Formula he Distance Formula he distance d between two points (, ) and (, ) is d = #( ) ( ). his lesson covers important geometric concepts in an algebraic contet. he Distance Formula, an etension of the uler ostulate to points in a twodimensional coordinate sstem, is an application of the thagorean heorem. he Midpoint Formula follows from the uler ostulate and the ide-plitter heorem. More Math ackground: p. D Lesson lanning and esources ee p. E for a list of the resources that support this lesson. ell inger ractice heck kills You ll Need For intervention, direct students to: kills Handbook, pp You will verif this formula in hapter 8. Lesson -8 he oordinate lane 5 pecial Needs L For Eample and Quick heck, have students check the results b graphing the segment and midpoint on a coordinate plane. elow Level L Use an overhead projector and right triangles on graph paper to relate the distance formula and the thagorean heorem. learning stle: tactile learning stle: visual 5

2 . each EXMLE Finding Distance Find the distance between (5, ) and (-, -) to the nearest tenth. Guided Instruction Math ip oint out that the term coordinate describing one number on a one-dimensional number line in Lesson -5 is epanded in this lesson to describe an ordered pair of two numbers on a two-dimensional coordinate plane. EXMLE eaching ip Have students sketch the relative positions of the points in the eample. Have them draw a horizontal line through (-, -) and a vertical line through (5, ) and draw segment to form a right triangle. sk: What rule relates the legs to the hpotenuse of a right triangle? thagorean heorem For: Distance ctivit Use: Interactive etbook, -8 North entral edar Quick heck Jackson (, ) mphon Let (5, ) be (, ) and (-, -) be (, ). d = #( ) ( ) d = #( 5) ( ) d = #(9) () d = "8 9 = "90 Use the Distance Formula. ubstitute. implif Use a calculator. o the nearest tenth, = 9.5. a. has endpoints (, -) and (-, ). Find to the nearest tenth. 8. b. ritical hinking In Eample, suppose ou let (-, -) be (, ) and (5, ) be (, ). Do ou get the same result? Wh? Yes; the differences are opposites, and the square of a number and the square of its opposite are the same. EXMLE eal-world onnection ravel Each morning Juanita takes the lue Line subwa from Oak tation to Jackson tation. s the map at the left shows, Oak tation is mile west and miles south of it laza. Jackson tation is miles east and miles north of it laza. Find the distance Juanita travels between Oak tation and Jackson tation. Let Oak(-, -) be (, ) and Jackson (, ) be (, ). EXMLE Error revention emind students to enclose negative numbers in parentheses when the substitute them into the Distance Formula. Have them double-check their work. Oak Elm it laza (0, 0) outh d = #( ) ( ) Use the Distance Formula. d = #( ()) ( ()) ubstitute. d = " implif. d = "9 = " Use a calculator. areers Mapmakers must report distances accuratel. heir work is complicated b the fact that Earth s surface is not a Euclidean plane. Quick heck Juanita travels about.7 miles between Oak tation and Jackson tation. a. Find the distance between Elm tation and mphon tation. about 8.9 mi b. Maple tation is located miles west and miles north of it laza. Find the distance between edar tation and Maple tation. about. mi Finding the Midpoint of a egment o find the coordinate of the midpoint of a segment on a number line, find the average or mean of the coordinates of the endpoints. he coordinate of the a a b midpoint of a segment with endpoints a and b is. You can etend this process (see net page) to find the coordinates of the midpoint of a segment in the coordinate plane. b 5 hapter ools of Geometr dvanced Learners L Encourage students to tr to derive the Midpoint Formula algebraicall b using the Distance Formula. English Language Learners ELL Make sure students understand that the Distance Formula gives a number (or measurement) while the Midpoint Formula gives an ordered pair. 5 learning stle: verbal learning stle: verbal

3 eal-world onnection well-made seesaw balances (without riders) at its midpoint. tud the diagram of with endpoints (, ) and (8, 5) (8, 5). is a horizontal segment and is a vertical segment. he coordinates of are (8, ). he coordinates of M, the midpoint of, are (, ). (, ) he coordinates of N, the midpoint of, are (8, ). (8, 5) vertical line through M and a horizontal line (, ) N through N meet at, the midpoint of. he coordinates of are (, ). (, ) M (8, ) hus, ou find the coordinates of the midpoint of a segment b averaging the -coordinates and averaging the -coordinates of the endpoints. dditional Eamples Find the distance between (-, ) and (, -) to the nearest tenth.. Use the diagram from Eample. How far is the subwa ride from Oak to mphon? ound to the nearest tenth..5 mi Guided Instruction Ke oncepts Formula he Midpoint Formula Quick heck Quick heck he coordinates of the midpoint M of with endpoints (, ) and (, ) are the following: EXMLE Finding the Midpoint lgebra Q has endpoints Q(, 5) and (7, -9). Find the coordinates of its midpoint M. Let (, 5) be (, ) and (7, -9) be (, ). -coordinate of M = = 7 = 0 = 5 5 (9) -coordinate of M = = = = - he coordinates of midpoint M are (5, -). Find the coordinates of the midpoint of XY with endpoints X(, -5) and Y(, ). (, ) EXMLE M Q, Finding an Endpoint lgebra he midpoint of is M(, ). One endpoint is (, ). Find the coordinates of the other endpoint. Use the Midpoint Formula. Let the coordinates of be (, ). = Midpoint Formula = =- + Multipl each side b. 8 =- + 9 = 0 = he coordinates of are (9, 0). - O he midpoint of XY has coordinates (, -). X has coordinates (, -). Find the coordinates of Y. (, 9) Lesson -8 he oordinate lane 55 8 O Q 8 M Visual Learners Have students use graph paper to find the midpoints M, N, and at the top of page 55. EXMLE EXMLE onnection to lgebra If necessar, review how to solve linear equations before beginning Eample. eaching ip Discuss with the class how the Distance and Midpoint Formulas provide was to solve geometr problems without having to use a drawing. dditional Eamples has endpoints (8, 9) and (-, -). Find the coordinates of its midpoint M. (, ) he midpoint of DG is M(-, 5). One endpoint is D(, ). Find the coordinates of the other endpoint G. (, ) esources Dail Notetaking Guide -8 L Dail Notetaking Guide -8 dapted Instruction L losure Use a coordinate plane to draw an obtuse triangle whose vertices have whole number coordinates. hen find the length and midpoint of each side. heck students work. 55

4 . ractice ssignment Guide -7, hallenge 57- EXEIE For more eercises, see Etra kill, Word roblem, and roof ractice. ractice and roblem olving ractice b Eample for Help Eample (page 5) Find the distance between the points to the nearest tenth.. J(, -), K(, 5). L(0, ), M(-8, ) 8. N(-, -), (-, -) 8. (0, ), (0, ) 9 5. (, ), D(-8, 8).. E(, -), F(-, ) 0 7. Q(, -), (5, ) 5 8. (0, 5), (, ). 9. X(-, -), Y(5, 5).0 est rep -9 Mied eview Homework Quick heck o check students understanding of ke skills and concepts, go over Eercises, 0, 7, 8, 50. Error revention! Eercises 9 emind students to substitute the four coordinates correctl into the Distance Formula. If students substitute incorrectl, have them write,,, and above the given coordinates. Eercise If necessar, review the arithmetic of mied numbers. Eercise sk: Wh is the point (0, 0) called the origin? ample: It is a starting point for numbering coordinates on the coordinate plane. G Enrichment Guided roblem olving eteaching dapted ractice ractice Name lass Date ractice -7 Find the area of each rectangle with the given base and height. erimeter, ircumference, and rea. base: ft. base: 0 in.. base: m height: in. height:.5 d height: 0 cm Find the circumference of each circle in terms of π Find the perimeter and area of each rectangle with the given base and height. 7. b = 7 cm, h = cm 8. b = cm, h = cm 9. b = in., h = 0.5 in. 0. b = 7 ft, h = ft. b = m, h = 9 m. b = m, h = 7 m.9 L L L L L Eample (page 5) Eample (page 55) Eample (page 55). 5.0; (.5, ). 5.8; (.5, 0.5). 7.; (.5, 0.5) 5. 5.; (.5, ). 0; (, ) 7..8; (, ) ppl Your kills nline Homework Help Visit: Hchool.com Web ode: aue-007 Use the map in Eample on page 5. Find the distance between the stations. 0. North and outh 9 mi. Oak and mphon about.5 mi. it laza and edar about. mi Use the map at the right. Find the distances between the stations to the nearest tenth.. tation and tation.. tation and tation located at (5, 8) 5.8 tation tation and tation D located at (, 0) 5.8. tation E at (, ) and tation F at (5, ) List the stations,, D, E, and F in the order of - tation least to greatest distance from tation.,, D, E, F lgebra Find the coordinates of the midpoint of HX. 8. H(0, 0), X(8, ) (, ) 9. H(-, ), X(7, -) (, ) 0. H(, 8), X(-, -) (.5, ). H(7, 0), X(5, -8) (, ). H(-., 5.), X(.8, -) (.5,.). H 7,,X, (8, ) lgebra he coordinates of point are given. he midpoint of has coordinates (5, 8). Find the coordinates of point.. (0, ) (0, 0) 5. (5, -5) (5, ). (0, 8) (0, ) 7. (-, 8) (, ) 8. (, ) (9, 8) 9. (.5, -.5) (5.5,.5) n endpoint and a midpoint are given. Find the coordinates of the other endpoint. 0. endpoint (, ), midpoint (5, ). endpoint (, ), midpoint (, -) (8, 8) (, ) Find (a) Q to the nearest tenth and (b) the coordinates of the midpoint of Q.. (, ), Q(, ). (0, -), Q(, ). (-, -), Q(, ) 5. (-5, ), Q(0, ). (-, -), Q(5, -7) 7. (-5, -), Q(-, -5) 8. (-, -5), Q(-, ) 9. (, ), Q(, -) 0. (, ), Q(, 0).7; (.5, ) 5.; (, 0.5).; (.5, ). he midpoint of is the origin. oint is located in Quadrant II. What quadrant contains point? IV. Graph the points (, ), (, -), (8, 7), and D(, 9). Draw quadrilateral D. Use the Midpoint Formula to find the midpoints of and D. What appears to be true? ee back of book. Find the perimeter and area of each figure. ll angles in the figures are right angles hapter ools of Geometr earson Education, Inc. ll rights reserved. Find the area of each circle in terms of π Find the area and perimeter of rectangle D with vertices (, 7), (9, 7), (9, -), and D(, -). 0. Find the perimeter of Q with vertices (-, 9), Q(7, -), and (-, -).. he circumference of a circle is p. Find the diameter and the radius. p 8 5

7. about 5 units; he dist. V is less than the dist. U, so V ± VU U ± VU. eal-world for Help For a guide to solving Eercise 7, see page 0. onnection eens average about 7 hours on the phone each week.

5 7. about 5 units; he dist. V is less than the dist. U, so V ± VU U ± VU. eal-world for Help For a guide to solving Eercise 7, see page 0. onnection eens average about 7 hours on the phone each week. hallenge. Multiple hoice (, -) and (-, ) are points in a coordinate plane. M is the midpoint of. What is the length of M to the nearest tenth of a unit?.5 units. units 7. units.0 units For each graph, find (a) to the nearest tenth and (b) the coordinates of the midpoint of O 9 O 8 O 9 9. units; (.5, 0) 0.8 units; (, ) 5. units; (, 0.5) 7. Writing n airplane at (80, 0) needs to fl to both U(0, 0) and V(0, 85). What is the shortest possible distance for the trip? Eplain. ee above left. 8. Navigation boat at X(5, -) needs to travel G to Y(-, 9) or Z(7, -). Which point is closer? What is the distance to the closer point? Z; about units ommunications he cell phone screen at the right shows coordinates of si cities from a grid placed on North merica b a long-distance carrier. he carrier finds distance b the Distance Formula. Each grid unit equals "0. mile. Find the distance between each pair of cities to the nearest mile. 9. Houston and hicago 9 mi 50. Denver and New Orleans 07 mi 5. oston and an Francisco 9 mi 5. New Orleans and Houston 8 mi an Francisco hicago New Orleans Denver Houston oston OUE: eter H. Dana (895, 870) (5985, 9) (88, 5) (790, 588) (89, 5) (, ) Graph X(, ), Y(, ), (, ), (0, ), and (, ). For each point described below, give two sets of possible coordinates if the eist. Otherwise, write eactl one point and give the coordinates, or not possible and eplain nswers ma * ) * ) * ) * ) var. amples 5. point D so that D XY 5. E so that E XY are given. (, ), (0,.5) * ) * ) (0, 0), (8, ) * ) * ) 55. point F so that F ' XY 5. point G so that G ' XY (, 0), (, ) * ) * ) * ) * ) (0, 0), (5, 0) 57. point H so that HX Y, and H XY eactl one pt., H ( 5, ) * ) * ) * ) * ) 58. point J so that J ' XY, and J ' Y eactl one pt., J (, ) 59. Open-Ended In a coordinate plane, draw an. Draw another segment that is both congruent and parallel to. Label the new segment D in such a wa that D is a quadrilateral. a f. ee margin. a. Find and D. What do ou notice? b. Write a conjecture that generalizes the result ou found in part (a). c. Find the midpoint of and the midpoint of D. What do ou notice? d. Write a conjecture that generalizes the result ou found in part (c). e. Find the midpoint E of D and the midpoint F of. Find EF and. What do ou notice? f. Write a conjecture that generalizes the result ou found in part (e). Lesson -8 he oordinate lane 57 Eercise ome students ma tr to find the midpoint and use it to calculate M. oint out that students can find and multipl it b because? 5 M. lternative Method Eercises fter students solve the eercises, ask: an ou echange points (, ) and (, ) and still have the correct lengths and midpoints? Help students recognize that order doesn t affect the squares of the differences in the Distance Formula and that the sums in the Midpoint Formula are the same because of the ommutative ropert of ddition. Eercises 5 5 tudents work informall with the properties of parallel and perpendicular lines in a coordinate plane. In hapter, these properties will be formall treated. 59.a f. nswers ma var. amples are given. a. D b. If two opp. sides of a quad. are both n and O, then the other two opp. sides are O. c. he midpts. are the same. d. If one pair of opp. sides of a quad. are both n and O, then its diagonals bisect each other. e. EF f. If a pair of opp. sides of a quad. are both n and O, then the segment joining the midpts. of the other two sides has the same length as each of the first pair of sides. 57

. ssess & eteach Lesson Quiz has coordinates (, 8). has coordinates (0, ). has coordinates ( 5, ).. Find the distance between and to the nearest tenth... Find to the nearest tenth. 5.. Find the midpoint M of to the nearest tenth.

6 . ssess & eteach Lesson Quiz has coordinates (, 8). has coordinates (0, ). has coordinates ( 5, ).. Find the distance between and to the nearest tenth... Find to the nearest tenth. 5.. Find the midpoint M of to the nearest tenth. (, ). is the midpoint of D. Find the coordinates of endpoint D. (, ) 5. n airplane flies from tanton to Mercur in a straight flight path. Mercur is 00 miles east and 00 miles south of tanton. How man miles is the flight? 500 mi. oni rides miles north, then 5 miles west, and then miles south. t the end of her ride, how far is oni from her starting point, measured in a straight line? mi Eercise 0 est rep Multiple hoice Geometr in Dimensions You can use three coordinates (,, z) to locate points in three dimensions. oint has coordinates (,.5, 9). 0. Give the coordinates of points,,, D, E, F, and G. ee margin.. Draw three aes like those shown. hen graph (, 5, 9). ee margin. Distance in Dimensions In a three-dimensional coordinate sstem, the distance between two points (,, z ) and (,, z ) can be found using this etension of the Distance Formula. d = #( ) ( ) (z z ) Find the distance between each pair of points to the nearest tenth.. (,, ), (-,, 9).5 units. Q(0,, 5), Y(-8, 0, ).7 units. What are the coordinates of the point that is halfwa between (, ) and (-, 8)?. (-9,.5). (-9,.5). (-8, 9) D. (,.5) 5. Which point lies the farthest from the origin? J F. (0, -7) G. (5, ) H. (-, -) J. (-, 8). segment has endpoints (, -8) and (, ). What are the coordinates of its midpoint? D. (9, 0). (-5, 0). (5, -0) D. (9, ) F G D z E lternative ssessment Have each student graph two points in a coordinate plane and then echange graphs with a partner. Each partner should calculate the length and the midpoint of the segment whose endpoints are the two graphed points. est rep esources For additional practice with a variet of test item formats: tandardized est rep, p. 75 est-aking trategies, p.70 est-aking trategies with ransparencies hort esponse 8. [] a. ( 8, 9) b. 7.9 [] one part correct 58 hapter ools of Geometr 7. usan takes the bus to visit her friend in Valle tream. he gets on a bus at Gibson and rides to Hewlett. t Hewlett, she transfers to a different bus and rides to Valle tream. he numbers on the grid represent miles. o the nearest tenth of a mile, how far does usan travel? J F. 5.8 miles G. 7. miles H. 9. miles J. 0.8 miles 8. he midpoint of is N(-, ). One endpoint is (0, -7). a. What are the coordinates of? b. What is the length of to the nearest tenth of a unit? 9. a. oints (-, ), Q(, ), and are collinear. One of the points is the midpoint of the segment formed b the other two points. What are the possible coordinates of? a b. ee margin, p. 59. b. Q =!0. Does this information affect our answer to part (a)? Eplain. Gibson O Hewlett Valle tream Woodlawn (0, 0, 0) (, 0, 0) (,.5, 0) D (0,.5, 0) E (0, 0, 9) F (, 0, 9) G (0,.5, 9). z

7 Mied eview for Help Lesson -7 Use a straightedge and compass ee margin. 70. Draw. onstruct Q so that Q =. 7. Draw LK. onstruct the perpendicular bisector of LK. Use this heckpoint Quiz to check students understanding of the skills and concepts of Lessons -5 through Draw an obtuse &. onstruct & so that m& = m&. 7. Draw an acute &. onstruct the bisector of &. esources Grab & Go heckpoint Quiz Lesson - 7. Name & two other was. 75. m&q = 0. What is m&q? l, l 50 Q Lesson lgebra he length of is 5. If = + 8 and = -, find the value of Find if the coordinate of is 5 and the coordinate of is lgebra is the midpoint of EF. Find EF E F heckpoint Quiz Lessons -5 through -8 Use the figure for Eercises.. If = + 5 and D = + 8, find. 7. If m&e = 5 and m&ed = 5, find m&d. 0. If m&f = 0, find m&fd. 0 * X ) lesson quiz, Hchool.com, Web ode: aua-008. is the perpendicular bisector of Q at M. a. What is m&m? 90 b. If ) QM = 0, what is Q? 0 5. ) is the bisector of &. Name two congruent angles. l Ol. O is the bisector of right &O. Find m&o. 5 Use a straightedge to draw three figures like the ones shown at the right. hen do each construction ee margin. 7. onstruct FG so that FG = D onstruct &HK so that m&hk = m&lk. has endpoints (, 5) and (, ). 9. Find to the nearest tenth.. units 0. Find the coordinates of the midpoint of. (,.5) F L Lesson -8 he oordinate lane 59 E K D D heckpoint Quiz 7. D F G 8. L H 9. [] a. ( 0, 8), (, 5), (8, ) b. Yes, must be ( 0, 8) so that Q "0. [] part (a) correct or plausible eplanation for part (b) Q L K K 59

Segments, Rays, Parallel Lines and Planes Q L R M. Segment AB. Endpoint. Ray YX. Naming Segments and Rays

Segments, Rays, Parallel Lines and Planes Q L R M. Segment AB. Endpoint. Ray YX. Naming Segments and Rays - egments, ays, arallel ines and lanes -. lan What You ll earn To identify segments and rays To recognize parallel lines... nd Why To identify compass directions that can be represented by opposite rays,