Coordinate Proof

Page 1 of 84.7What you should learnGOAL 1 Place geometricfigures in a coordinate plane.GOAL 2Write a coordinateproof.Triangles andCoordinate ProofGOAL 1PLACING FIGURES IN A COORDINATE PLANESo far, you have studied two-column proofs, paragraph proofs, and flow proofs. Acoordinate proof involves placing geometric figures in a coordinate plane. Thenyou can use the Distance Formula and the Midpoint Formula, as well as postulatesand theorems, to prove statements about the figures.Why you should learn itACTIVITYSometimes a coordinateproof is the most efficientway to prove a statement.DevelopingConceptsPlacing Figures in a Coordinate Plane1Draw a right triangle with legs of 3 unitsand 4 units on a piece of grid paper. Cutout the triangle.2Use another piece of grid paper to draw acoordinate plane.3INTNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor extra examples.11Sketch different ways that the triangle canbe placed on the coordinate plane. Which ofthe ways that you placed the triangle is bestfor finding the length of the hypotenuse?xPlacing a Rectangle in a Coordinate PlaneEXAMPLE 1STUDENT HELPyPlace a 2-unit by 6-unit rectangle in a coordinate plane.SOLUTIONChoose a placement that makes finding distances easy. Here are two possibleplacements.yy(0, 6)(2, 6)( 3, 2)(3, 2)12( 3, 0)(0, 0)1(3, 0)x(2, 0)1xOne vertex is at the origin, andthree of the vertices have at leastone coordinate that is 0.One side is centered at the origin,and the x-coordinates areopposites.4.7 Triangles and Coordinate Proof243

Page 2 of 8Once a figure has been placed in a coordinate plane, you can use the DistanceFormula or the Midpoint Formula to measure distances or locate points.xyUsingAlgebraUsing the Distance FormulaEXAMPLE 2A right triangle has legs of 5 units and 12 units. Place the triangle in a coordinateplane. Label the coordinates of the vertices and find the length of the hypotenuse.SOLUTIONy(12, 5)One possible placement is shown.Notice that one leg is vertical and theother leg is horizontal, which assuresthat the legs meet at right angles. Pointson the same vertical segment have thesame x-coordinate, and points on thesame horizontal segment have the samey-coordinate.d13(0, 0)(12, 0) xYou can use the Distance Formula to find the length of the hypotenuse.22d (x 2 º x ( y2 º y 1) 1) Distance Formula (1 2 º 0 )2 (5 º 0 )2Substitute. 1 6 9 Simplify. 13Evaluate square root.Using the Midpoint FormulaEXAMPLE 3In the diagram, MLO KLO.y M (0, 160)Find the coordinates of point L.SOLUTIONLBecause the triangles are congruent, itÆÆfollows that ML KL . So, point L mustÆbe the midpoint of MK . This means youcan use the Midpoint Formula to find thecoordinates of point L. x x2y y21212L(x, y) , 1602 0 0 2160 24420OMidpoint Formula , Substitute. (80, 80)Simplify.The coordinates of L are (80, 80).Chapter 4 Congruent TrianglesK (160, 0)20x

Page 3 of 8GOAL 2WRITING COORDINATE PROOFSOnce a figure is placed in a coordinate plane, you may be able to provestatements about the figure.EXAMPLE 4ProofWriting a Plan for a Coordinate ProofÆ Write a plan to prove that SO bisects PSR.yS (0, 4)GIVEN Coordinates of vertices of POS and ROSÆ PROVE SO bisects PSR1P ( 3, 0)O (0, 0) R (3, 0)xSOLUTIONPlan for Proof Use the Distance Formula to find the side lengths of POS and ROS. Then use the SSS Congruence Postulate to show that POS ROS.Finally, use the fact that corresponding parts of congruent triangles are congruentÆ to conclude that PSO RSO, which implies that SO bisects PSR.The coordinate proof in Example 4 applies toa specific triangle. When you want to prove astatement about a more general set of figures,it is helpful to use variables as coordinates.For instance, you can use variable coordinatesto duplicate the proof in Example 4. OnceÆ this is done, you can conclude that SO bisects PSR for any triangle whose coordinates fitthe given pattern.EXAMPLE 5yS (0, k)P ( h, 0)O (0, 0) R (h, 0)xUsing Variables as CoordinatesRight OBC has leg lengths of h units andk units. You can find the coordinates of pointsB and C by considering how the triangle isplaced in the coordinate plane.Point B is h units horizontally from the origin,so its coordinates are (h, 0). Point C is h unitshorizontally from the origin and k units verticallyfrom the origin, so its coordinates are (h, k).yC (h, k)k unitsO (0, 0)B (h, 0)h unitsxINTSTUDENT HELPNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor extra examples.ÆYou can use the Distance Formula to find the length of the hypotenuse OC.OC (h º 0 )2 (k º 0 )2 h 2 k2 4.7 Triangles and Coordinate Proof245

Page 4 of 8Writing a Coordinate ProofEXAMPLE 6ProofGIVEN Coordinates of figure OTUVyPROVE OTU UVOU (m h, k )T (m, k )SOLUTIONÆCOORDINATE PROOF Segments OV andÆUT have the same length.O (0, 0)OV (h º 0 ) (0 º 0 ) h2xV (h, 0)2UT (m h º m )2 (k º k )2 hÆÆHorizontal segments UT and OV each have a slope of 0, which implies thatÆÆÆthey are parallel. Segment OU intersects UT and OV to form congruent alternateÆÆinterior angles TUO and VOU. Because OU OU, you can apply theSAS Congruence Postulate to conclude that OTU UVO.GUIDED PRACTICEVocabulary Check 1. Prior to this section, you have studied two-column proofs, paragraph proofs,and flow proofs. How is a coordinate proof different from these other typesof proof? How is it the same?Concept Check same right triangle in a coordinateplane are shown. Which placementis more convenient for findingthe side lengths? Explain yourthinking. Then sketch a thirdplacement that also makes itconvenient to find the side lengths.Skill Check y2. Two different ways to place theyCBABACxx3. A right triangle with legs of 7 units and 4 units has one vertex at (0, 0) andanother at (0, 7). Give possible coordinates of the third vertex.DEVELOPING PROOF Describe a plan for the proof.Æ 4. GIVEN GJ bisects OGH.PROVE GJO GJHy5. GIVEN Coordinates ofvertices of ABCPROVE ABC is isosceles.yGA (0, k )J1O2461Chapter 4 Congruent TrianglesHxC ( h, 0)B (h, 0) x

Page 5 of 8PRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 810.PLACING FIGURES IN A COORDINATE PLANE Place the figure in a coordinateplane. Label the vertices and give the coordinates of each vertex.6. A 5-unit by 8-unit rectangle with one vertex at (0, 0)7. An 8-unit by 6-unit rectangle with one vertex at (0, º4)8. A square with side length s and one vertex at (s, 0)CHOOSING A GOOD PLACEMENT Place the figure in a coordinate plane.Label the vertices and give the coordinates of each vertex. Explain theadvantages of your placement.9. A right triangle with legs of 3 units and 8 units10. An isosceles right triangle with legs of 20 units11. A rectangle with length h and width kFINDING AND USING COORDINATESIn the diagram, ABC is isosceles. Itsbase is 60 units and its height is 50 units.yB12. Give the coordinates of points B and C.13. Find the length of a leg of ABC.10Round your answer to the nearesthundredth.10A( 30, 0)xCUSING THE DISTANCE FORMULA Place the figure in a coordinate planeand find the given information.14. A right triangle with legs of 7 and 9 units; find the length of the hypotenuse.15. A rectangle with length 5 units and width 4 units; find the length of a diagonal.16. An isosceles right triangle with legs of 3 units; find the length of the hypotenuse.17. A 3-unit by 3-unit square; find the length of a diagonal.USING THE MIDPOINT FORMULA Use the given information and diagramto find the coordinates of H.18. FOH FJHy19. OCH HNMyJ (80, 80)M (90, 70)STUDENT HELPHOMEWORK HELPExample 1:Example 2:Example 3:Example 4:Example 5:Example 6:Exs. 6–11Exs. 12–17Exs. 18, 19Exs. 20, 21Exs. 22–25Exs. 26, 27HHN (90, 35)1010O(0, 0)40F (80, 0) xO(0, 0)C (45, 0)804.7 Triangles and Coordinate Proofx247

Page 6 of 8DEVELOPING PROOF Write a plan for a proof.ÆÆÆ20. GIVEN OS fi RT21. GIVEN G is the midpoint of HF.Æ PROVE OS bisects TOR.PROVE GHJ GFOyyR (0, 60)H (2, 6)SG110O(0, 0)J (6, 6)T (60, 0) x10xF (4, 0)1O(0, 0)USING VARIABLES AS COORDINATES Find the coordinates of anyunlabeled points. Then find the requested information.22. Find MP.23. Find OE.yyh unitsMNEk unitsh unitsO(0, 0)Px24. Find ON and MN.yNk unitsD (h, 0)x25. Find OT.yO (0, 0)F2h unitsO(0, 0)M (2h, 0) xOSTR2k unitsxUCOORDINATE PROOF Write a coordinate proof.26. GIVEN Coordinates of27. GIVEN Coordinates of OBC NPO and NMOPROVE NPO NMOand EDCPROVE OBC EDCyyD (h, 2k)P (0, 2h)E (2h, 2k)N (h, h)C (h, k)O (0, 0)248Chapter 4 Congruent TrianglesM (2h, 0) xO (0, 0)B (h, 0)x

Page 7 of 8PLANT STAND You buy a tall, three-legged28.yplant stand. When you place a plant on thestand, the stand appears to be unstable under theweight of the plant. The diagram at the rightshows a coordinate plane superimposed onone pair of the plant stand’s legs. The legs areextended to form OBC. Is OBC an isoscelestriangle? Explain why the plant stand may beunstable.B (12, 48)6xC (18, 0)6O(0, 0)TECHNOLOGY Use geometry software for Exercises 29–31. Follow thesteps below to construct ABC. Create a pair of axes. Construct point Aon the y-axis so that the y-coordinate ispositive. Construct point B on the x-axis.AConstruct a circle with a center at theorigin that contains point B. Label theother point where the circle intersectsthe x-axis C.CB Connect points A, B, and C to form ABC.Find the coordinates of each vertex.29. What type of triangle does ABC appear to be? Does your answer change ifyou drag point A? If you drag point B?30. Measure and compare AB and AC. What happens to these lengths as youdrag point A? What happens as you drag point B?31. Look back at the proof described in Exercise 5 on page 246. How does thatproof help explain your answers to Exercises 29 and 30?TestPreparation32. MULTIPLE CHOICE A square with side length 4 has one vertex at (0, 2).Which of the points below could be a vertex of the square?A¡(0, º2)B¡(2, º2)C¡D¡(0, 0)(2, 2)33. MULTIPLE CHOICE A rectangle with side lengths 2h and k has one vertex at(ºh, k). Which of the points below could not be a vertex of the rectangle?A¡ Challenge34.B¡(0, k)(ºh, 0)C¡D¡(h, k)(h, 0)yCOORDINATE PROOF Use theA (0, 2k)diagram and the given informationto write a proof.GIVEN Coordinates of DEA,HGÆEXTRA CHALLENGEwww.mcdougallittell.comH is the midpoint of DA,ÆG is the midpoint of EA.ÆÆPROVE DG EHD( 2h, 0)O(0, 0)E(2h, 0)4.7 Triangles and Coordinate Proofx249

Page 8 of 8MIXED REVIEWÆ xy USING ALGEBRA In the diagram, GRbisectsR CGF. (Review 1.5 for 5.1)35. Find the value of x.(4x 55) 36. Find m CGF.C15x FGPERPENDICULAR LINES AND SEGMENT BISECTORS Use the diagram todetermine whether the statement is true or false. (Review 1.5, 2.2 for 5.1) 37. PQ is perpendicular to LN .P38. Points L, Q, and N are collinear. Æ39. PQ bisects LN .MLNq40. LMQ and PMN are supplementary.WRITING STATEMENTS Let p be “two triangles are congruent” and let q be“the corresponding angles of the triangles are congruent.” Write thesymbolic statement in words. Decide whether the statement is true.(Review 2.3)41. p q42. q p43. p qQUIZ 3Self-Test for Lessons 4.5–4.7PROOF Write a two-column proof or a paragraph proof.(Lessons 4.5 and 4.6)ÆÆÆÆÆ1. GIVEN DF DG,ÆED HDÆÆSU TVPROVE EFD HGDFÆ2. GIVEN ST UT VU,PROVE STU TUVVTGDE3.HSUCOORDINATE PROOF Write a planfor a coordinate proof. (Lesson 4.7)yP(3, 4)M(8, 4)GIVEN Coordinates of vertices of OPM and ONMPROVE OPM and ONM arecongruent isosceles triangles.250Chapter 4 Congruent Triangles1O(0, 0) 3N (5, 0)x

figures in a coordinate plane. Write a coordinate proof. Sometimes a coordinate proof is the most efficient way to prove a statement. Why you should learn it GOAL 2 GOAL 1 What you should learn 4.7 Placing Figures in a Coordinate Plane Draw a right triangle with legs of 3 units and 4 units on a piece of grid paper. Cut out the triangle. Use .