12.8 Coordinate Proofs

12.8 Coordinate ProofsEssential QuestionHow can you use a coordinate plane to writea proof?Writing a Coordinate ProofWork with a partner.a. Use dynamic geometry—software to draw ABwith endpoints A(0, 0)and B(6, 0).b. Draw the vertical linex 3.c. Draw ABC so thatC lies on the line x 3.d. Use your drawing toprove that ABC isan isosceles triangle.Sample43C21B0A0132456PointsA(0, 0)B(6, 0)C(3, y)SegmentsAB 6Linex 3 1Writing a Coordinate ProofWork with a partner.— with endpoints A(0, 0) and B(6, 0).a. Use dynamic geometry software to draw ABb. Draw the vertical line x 3.c. Plot the point C(3, 3) and draw ABC. Then use your drawing to prove that ABCis an isosceles right triangle.Sample4PointsA(0, 0)B(6, 0)C(3, 3)SegmentsAB 6BC 4.24AC 4.24Linex 3C32CRITIQUINGTHE REASONINGOF OTHERSTo be proficient in math,you need to understandand use stated assumptions,definitions, and previouslyestablished results.1B0A0123456 1d. Change the coordinates of C so that C lies below the x-axis and ABC is anisosceles right triangle.e. Write a coordinate proof to show that if C lies on the line x 3 and ABC is anisosceles right triangle, then C must be the point (3, 3) or the point found in part (d).Communicate Your Answer3. How can you use a coordinate plane to write a proof?4. Write a —coordinate proof to prove that ABC with vertices A(0, 0), B(6, 0), andC(3, 3 3 ) is an equilateral triangle.Section 12.8int math1 pe 1208.indd 639Coordinate Proofs6391/29/15 4:58 PM

12.8 LessonWhat You Will LearnPlace figures in a coordinate plane.Core VocabulVocabularylarrycoordinate proof, p. 640Write coordinate proofs.Placing Figures in a Coordinate PlaneA coordinate proof involves placing geometric figures in a coordinate plane. Whenyou use variables to represent the coordinates of a figure in a coordinate proof, theresults are true for all figures of that type.Placing a Figure in a Coordinate PlanePlace each figure in a coordinate plane in a way that is convenient for finding sidelengths. Assign coordinates to each vertex.a. a rectangleb. a scalene triangleSOLUTIONIt is easy to find lengths of horizontal and vertical segments and distances from (0, 0),so place one vertex at the origin and one or more sides on an axis.a. Let h represent the length andk represent the width.b. Notice that you need to usethree different variables.yy(0, k)(h, k)(f, g)k(0, 0)(h, 0) xhMonitoring Progress(d, 0) x(0, 0)Help in English and Spanish at BigIdeasMath.com1. Show another way to place the rectangle in Example 1 part (a) that is convenientfor finding side lengths. Assign new coordinates.2. A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.Once a figure is placed in a coordinate plane, you may be able to prove statementsabout the figure.Writing a Plan for a Coordinate ProofWrite a plan to prove that ⃗SO bisects PSR.yS(0, 4)Given Coordinates of vertices of POS and ROSProve ⃗SO bisects PSR.2P( 3, 0)R(3, 0)O(0, 0) 4 xSOLUTIONPlan for Proof Use the Distance Formula to find the side lengths of POS and ROS.Then use the SSS Congruence Theorem to show that POS ROS. Finally, usethe fact that corresponding parts of congruent triangles are congruent to conclude thatSO bisects PSR. PSO RSO, which implies that ⃗640Chapter 12int math1 pe 1208.indd 640Congruent Triangles1/29/15 4:58 PM

Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com3. Write a plan for the proof.Given ⃗GJ bisects OGH.Prove GJO GJH6yG42JO2H x4The coordinate proof in Example 2 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.yS(0, k)For instance, you can use variable coordinates toduplicate the proof in Example 2. Once this isdone, you can conclude that ⃗SO bisects PSRfor any triangle whose coordinates fit thegiven pattern.R(h, 0)P( h, 0)O(0, 0)xApplying Variable CoordinatesPlace an isosceles right triangle in a coordinate plane. Then find the length of thehypotenuse and the coordinates of its midpoint M.SOLUTIONPlace PQO with the right angle at the origin. Let the length of the legs be k. Thenthe vertices are located at P(0, k), Q(k, 0), and O(0, 0).FINDING ANENTRY POINTAnother way to solveExample 3 is to place atriangle with point C at(0, h) on the y-axis and— on thehypotenuse ABx-axis. To make ACB aright angle, position A and— and CB—B so that legs CAhave slopes of 1 and 1,respectively.Slope is 1.Slope is 1.C(0, h)yyP(0, k)MO(0, 0)Q(k, 0) xUse the Distance Formula to find PQ, the length of the hypotenuse.—————PQ (k 0)2 (0 k)2 k2 ( k)2 k2 k2 2k2Use the Midpoint Formula to find the midpoint M of the hypotenuse.0 k k 0k kM —, — M —, —222 2() ( )—So, the length of the hypotenuse is 2k2 and the midpoint of the hypotenuse is( , ).k k2 2— —B(h, 0) xA( h, 0)Length of hypotenuse 2h() h h 0 0M —, — M(0, 0)22Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com4. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJ a right triangle? Find theside lengths and the coordinates of the midpoint of each side.Section 12.8int math1 pe 1208.indd 641Coordinate Proofs6411/29/15 4:58 PM

Writing Coordinate ProofsWriting a Coordinate ProofWrite a coordinate proof.yGivenCoordinates of vertices ofquadrilateral OTUVT(m, k)Prove OTU UVOU(m h, k)SOLUTION— and UT— have the same length.Segments OVO(0, 0)V(h, 0)xOV h 0 hUT (m h) m h— and OV— each have a slope of 0, which implies that they areHorizontal segments UT— intersects UT— and OV— to form congruent alternate interior angles,parallel. Segment OU— OU—. TUO and VOU. By the Reflexive Property of Congruence, OUSo, you can apply the SAS Congruence Theorem to conclude that OTU UVO.Writing a Coordinate ProofYou buy a tall, three-legged plant stand. Whenyou place a plant on the stand, the stand appearsto be unstable under the weight of the plant. Thediagram at the right shows a coordinate planesuperimposed on one pair of the plant stand’s legs.The legs are extended to form OBC. Prove that OBC is a scalene triangle. Explain why the plantstand may be unstable.SOLUTIONB(12, 48)362412O(0, 0)First, find the side lengths of OBC.——y4812C(18, 0)x—OB (48 0)2 (12 0)2 2448 49.5———BC (18 12)2 (0 48)2 2340 48.4OC 18 0 18Because OBC has no congruent sides, OBC is a scalene triangle by definition.— is longer than BC—, so the plant standThe plant stand may be unstable because OBis leaning to the right.Monitoring Progress5. Write a coordinate proof.GivenCoordinates of verticesof NPO and NMOProve NPO NMOHelp in English and Spanish at BigIdeasMath.comyP(0, 2h)N(h, h)O(0, 0)642Chapter 12int math1 pe 1208.indd 642M(2h, 0) xCongruent Triangles1/29/15 4:58 PM

12.8 ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY How is a coordinate proof different from other types of proofs you have studied?How is it the same?2. WRITING Explain why it is convenient to place a right triangle onythe grid as shown when writing a coordinate proof.(0, b)(a, 0) x(0, 0)Monitoring Progress and Modeling with MathematicsIn Exercises 3–6, place the figure in a coordinateplane in a convenient way. Assign coordinates to eachvertex. Explain the advantages of your placement.(See Example 1.)In Exercises 9–12, place the figure in a coordinate planeand find the indicated length.9. a right triangle with leg lengths of 7 and 9 units;Find the length of the hypotenuse.3. a right triangle with leg lengths of 3 units and 2 units10. an isosceles triangle with a base length of 60 units and4. a square with a side length of 3 units5. an isosceles right triangle with leg length pa height of 50 units; Find the length of one of the legs.11. a rectangle with a length of 5 units and a width of4 units; Find the length of the diagonal.6. a scalene triangle with one side length of 2m12. a square with side length n; Find the length ofIn Exercises 7 and 8, write a plan for the proof.(See Example 2.)7. GivenProveCoordinates of vertices of OPM and ONM OPM and ONM are isosceles triangles.yP(3, 4)4M(8, 4)O(0, 0)ProveIn Exercises 13 and 14, graph the triangle with the givenvertices. Find the length and the slope of each side ofthe triangle. Then find the coordinates of the midpointof each side. Is the triangle a right triangle? isosceles?Explain. (Assume all variables are positive and m n.)(See Example 3.)13. A(0, 0), B(h, h), C(2h, 0)28. Giventhe diagonal.N(5, 0) 8 x14. D(0, n), E(m, n), F(m, 0)—.G is the midpoint of HFIn Exercises 15 and 16, find the coordinates of anyunlabeled vertices. Then find the indicated length(s). GHJ GFO15. Find ON and MN.yy4H(1, 4)J(6, 4)yNO(0, 0) D(h, 0) M(2h, 0) xO(0, 0)24TO2k unitsUxF(5, 0) xSection 12.8int math1 pe 1208.indd 643SRk unitsG216. Find OT.Coordinate Proofs6431/29/15 4:58 PM

PROOF In Exercises 17 and 18, write a coordinate proof.(See Example 4.)17. GivenProveCoordinates of vertices of DEC and BOC DEC BOCyD(h, 2k)B(h, 0)D — —()( w2 , 2v )xA (h, k) B ( h, 0) C (2h, 0) D (2h, k) 24. THOUGHT PROVOKING Choose one of the theoremsyou have encountered up to this point that you thinkwould be easier to prove with a coordinate proof thanwith another type of proof. Explain your reasoning.Then write a coordinate proof.yA(0, 2k)GO E(2h, 0) x25. CRITICAL THINKING The coordinates of a triangleare (5d, 5d ), (0, 5d ), and (5d, 0). How should thecoordinates be changed to make a coordinate proofeasier to complete?19. MODELING WITH MATHEMATICS You and yourcousin are camping in the woods. You hike to apoint that is 500 meters east and 1200 meters northof the campsite. Your cousin hikes to a point that is1000 meters east of the campsite. Use a coordinateproof to prove that the triangle formed by yourposition, your cousin’s position, and the campsite isisosceles. (See Example 5.)26. HOW DO YOU SEE IT? Without performing anycalculations, how do you know that the diagonalsof square TUVW are perpendicular to each other?How can you use a similar diagram to show thatthe diagonals of any square are perpendicular toeach other?3yT(0, 2)W( 2, 0)U(2, 0) 420. MAKING AN ARGUMENT Two friends see a drawingof quadrilateral PQRS with vertices P(0, 2), Q(3, 4),R(1, 5), and S( 2, 1). One friend says thequadrilateral is a parallelogram but not a rectangle.The other friend says the quadrilateral is a rectangle.Which friend is correct? Use a coordinate proof tosupport your answer.21. MATHEMATICAL CONNECTIONS Write an algebraicexpression for the coordinates of each endpoint of aline segment whose midpoint is the origin. 3Chapter 12int math1 pe 1208.indd 644xV(0, 2)27. PROOF Write a coordinate proof for each statement.a. The midpoint of the hypotenuse of a right triangleis the same distance from each vertex of thetriangle.Reviewing what you learned in previous grades and lessons ⃗YW bisects XYZ such that m XYW (3x 7) and m WYZ (2x 1) .28. Find the value of x.4b. Any two congruent right isosceles triangles can becombined to form a single isosceles triangle.Maintaining Mathematical Proficiency644——width of k has a vertex at ( h, k). Which point cannotbe a vertex of the rectangle?Coordinates of DEA, H is the midpoint—, G is the midpoint of EA—.of DA— EH—Prove DGD( 2h, 0)( )( w2 , 2v )23. REASONING A rectangle with a length of 3h and a18. GivenH(w, 0), (0, v), ( w, 0), and (0, v). What is themidpoint of the side in Quadrant III?w vw v—, —AB —, — 2 22 2C E(2h, 2k)C(h, k)O(0, 0)22. REASONING The vertices of a parallelogram are(Section 8.5)29. Find m XYZ.Congruent Triangles1/29/15 4:58 PM

Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .