Algebra Give the coordinates of B without using any new variables. x 2. New Vocabulary midsegment of a trapezoid EXAMPLE

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1 6-7. lan 6-7 roofs Using Coordinate eometr EB jetives To prove theorems using figures in the oordinate plane Eamples lanning a Coordinate eometr roof eal-world Connetion What You ll earn To prove theorems using figures in the oordinate plane... nd Wh To use oordinate geometr to prove that a flag design inludes a rhomus, as in Eample Chek Skills You ll eed for Help esson 6-6. raph the rhomus with verties (, ), B(7, ), C(4, -), and D(-, -). Then, onnet the midpoints of onseutive sides to form a quadrilateral. What do ou notie aout the quadrilateral? The quad. is a retangle. lgera ive the oordinates of B without using an new variales.. retangle 3. isoseles triangle (a, ) C(0, ) B ( a, 0) C(0, ) ath Bakground (a, 0) B (a, 0) The Trapezoid idsegment Theorem is an etension of the Triangle idsegment Theorem. This eomes lear suessivel dereasing the shorter ase of a trapezoid until it measures 0. ew Voaular midsegment of a trapezoid Building roofs in the Coordinate lane ore ath Bakground: p. 86D esson lanning and esoures See p. 86E for a list of the resoures that support this lesson. oweroint Bell inger ratie Chek Skills You ll eed For intervention, diret students to: Finding idpoints esson -8: Eample 3 Etra Skills, Word rolems, roof ratie, Ch. Finding issing Coordinates esson 6-6: Eamples, Etra Skills, Word rolems, roof ratie, Ch. 6 In esson 5-, ou learned aout midsegments of triangles. trapezoid also has a midsegment. The midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel opposite sides. It has two unique properties. Ke Conepts Theorem 6-8 Trapezoid idsegment Theorem () The midsegment of a trapezoid is parallel to the ases. () The length of the midsegment of a trapezoid is half the sum of the lengths of the ases. 6 T, 6, and = (T + ). Formulas for slope, midpoint, and distane are used in a proof of Theorem 6-8. EXE lanning a Coordinate eometr roof Developing roof lan a oordinate proof of Theorem 6-8. iven: is the midsegment of trapezoid T. 8. rove: 6 T, 6, and = (T + ). (, ) (d, ) lan: lae the trapezoid in the oordinate plane with a verte at the origin and a ase along the -ais. Sine midpoints will e involved, use multiples of to name oordinates. To show lines are parallel, hek for equal slopes. To ompare lengths, use the Distane Formula. T (a, 0) T 348 Chapter 6 Quadrilaterals 348 Speial eeds elate the Trapezoid idsegment Theorem to the Triangle idsegment Theorem from esson 5-. Students should reognize that eah midsegment is parallel to its ase(s) and half the length of the ase (or sum of ases). learning stle: veral Below evel Before disussing the proofs, review the formulas for distane, midpoint, and slope. learning stle: veral

2 Quik Chek Complete the oordinate proof of Theorem 6-8. See margin, p 350. a. Find the oordinates of midpoints and. How do the multiples of help?. Find and ompare the slopes of, T, and.. Find and ompare the lengths, T, and. d. In parts () and (), how does plaing a ase along the -ais help?. Teah uided Instrution eal-world Quik Chek ratie Eample Eample (page 348) EXE eal-world Connetion lgera The retangular flag at the left is onstruted onneting the midpoints of its sides. Use oordinate geometr to prove that the quadrilateral formed onneting the midpoints of the sides of a retangle is a rhomus. iven: is a retangle. T, W, V, U are midpoints of its sides. rove: TWVU is a rhomus. lan: lae the retangle in the oordinate plane with two sides along the aes. Use multiples of to name oordinates. (0, ) T W U (a, ) V (a, 0) rhomus is a parallelogram with four ongruent sides. From esson 6-6, Eample, ou know that TWVU is a parallelogram. To show TW > WV > VU > UT, use the Distane Formula. Coordinate roof: B the idpoint Formula, the oordinates of the midpoints are T(0,), W(a, ), V(a, ), and U(a, 0). B the Distane Formula, TW = Î (a 0) ( ) = Î a WV = Î (a a) ( ) = Î a VU = Î (a a) (0 ) = Î a UT = Î (0 a) ( 0) = Î a TW > WV> VU > UT, so parallelogram TWVU is a rhomus. Critial Thinking Eplain wh the proof using (0, ), (a, ), (a, 0), and (0, 0) is easier than a proof using (0, ), (a, ), (a, 0), and (0, 0). See ak of ook. EXECISES For more eerises, see Etra Skill, Word rolem, and roof ratie. ratie and rolem Solving Connetion flag s length, alled the fl, usuall is greater than its width, alled the hoist. for Help.a. W( e ); Z( ) a, d,. W(a, ); Z( ± e, d). W(a, ); Z( ± e, d) d. ; it uses multiples of to name the oordinates of W and Z.. W and Z are the midpoints of and ST, respetivel. In parts (a) (), find the oordinates of W and Z. a d. See left. a... (a, ) S(, d) (4a, 4) S(4, 4d) S(, d) (a, ) W Z W Z W Z (?,?) (?,?) (?,?) T(e, 0) T(e, 0) T(4e, 0) d. You are to plan a oordinate proof involving the midpoint of WZ. Whih of the figures (a) () would ou prefer to use? Eplain. esson 6-7 roofs Using Coordinate eometr 349 EXE Connetion to lgera Help students understand that a is a valid oordinate enouraging them to tr the proof using other oordinates. The will see that the hoie of variale is aritrar. oweroint dditional Eamples Eamine trapezoid T. Eplain wh ou an assign the same -oordinate to points and. Sine T n and T is horizontal, is horizontal. Use oordinate geometr to prove that the quadrilateral formed onneting the midpoints of rhomus BCD is a retangle. C( a, 0) Sample method: Show that diagonals are ongruent. esoures Dail otetaking uide Dail otetaking uide 6-7 dapted Instrution Closure (a, 0) How are the midsegments of trapezoids and triangles alike? How are the different? Both are parallel to ases; triangle midsegments are half the length of the third side, ut trapezoid midsegments are the average length of oth ases. D(0, ) B(0, ) dvaned earners 4 fter reading Eample, students should e ale to prove that the quadrilateral formed onneting the midpoints of a square is also a square. learning stle: veral English anguage earners E Define the midsegment of a trapezoid. sk: How else ould the midsegment have een defined? joining the midpoints of the two ases Wh might this not e its definition? resulting midsegment is not parallel to the legs learning stle: veral 349

3 3. ratie ssignment uide B -36 C Challenge 37-4 Test rep 4-45 ied eview Homework Quik Chek To hek students understanding of ke skills and onepts, go over Eerises 3, 6, 0,, 6. Error revention! Eerise Help students oserve that onl (a) requires the use of frations. Eerises, 3 These eerises plan oordinate proofs for theorems alread proved geometri methods. s a lass, disuss the similarities and differenes etween the methods. oint out that eah new theorem and method of proof provides more resoures to use in proving new theorems. S uided rolem Solving Enrihment eteahing dapted ratie ratie earson Eduation, In. ll rights reserved. ame Class Date ratie 6-7. iven H with perpendiular isetors i,, and m, omplete the following to show that i,, and m interset in a point. q a. The slope of H is p. What is the slope of line i?. The midpoint of H is (p, q). Show that the equation of line i is p p = + q - q q.. The midpoint of H is (r + p, 0). What is the equation of line m? rp d. Show that lines i and m interset at (r + p, q + q). q e. The slope of is r. What is the slope of line? f. What is the midpoint of? r g. Show that the equation of line is = q + q - r q. rp h. Show that lines and m interset at (r + p, q + q). i. ive the oordinates for the point of intersetion of i,, and m. Complete Eerises and 3 without using an new variales.. HC is a rhomus. a. Determine the oordinates of.. Determine the oordinates of H.. Find the midpoint of H. d. Find the slope of H. 3. DFS is a kite. a. Determine the oordinates of S.. Find the midpoint of S.. Find the slope of S. d. Find the midpoint of DF. e. Find the slope of DF. 4. Complete the oordinates for retangle DHC. Then use oordinate geometr to prove the following statement: The diagonals of a retangle are ongruent (Theorem 6-). iven: retangle DHC rove: DC H S H ( a, ) roofs Using Coordinate eometr (0, 6a) H (p, 0) C (0, 0) D (4a, 0) F (0, a) i m (0, q) D H (a, ) (0, 0) C (r, 0) rolem Solving Hint When ou read large loks of math tet, over all ut a few lines to help ou fous. a. origin. -ais. d. oordinates 3a. -ais. Distane 4a. rt. l. legs. multiples of d. e. f. idpoint g. Distane Eample (page 349) Developing roof Complete the plan for eah oordinate proof.. The diagonals of a parallelogram iset eah other (Theorem 6-3). iven: arallelogram BCD rove: C isets BD, and BD isets C. B(, ) C(a +, ) lan: lae the parallelogram in the E oordinate plane with a verte at the a. 9 and a side along the. 9. D(a, 0) Sine midpoints will e involved, use multiples of. 9 to name oordinates. To show segments iset eah other, show the midpoints have the same d The diagonals of an isoseles trapezoid are ongruent (Theorem 6-6). iven: Trapezoid EFH with FE > H F(-, ) (, ) rove: E > HF lan: The trapezoid is isoseles, so plae one ase on the -ais so that the a. 9 isets its ases. To show the diagonals are ongruent, use the. 9 Formula. E(-a, 0) H(a, 0) 4. The median to the hpotenuse of a right triangle is half the hpotenuse. iven: # is a right triangle with right &. is the midpoint of. rove: = (0, a) lan: lae the right triangle in the oordinate plane with the verte of the a. 9 at the origin and the (, 0). 9 along eah ais. Sine midpoints will e involved, use. 9 to name oordinates for points d. 9 and e. 9. Use the f. 9 Formula to find the oordinates of. To ompare lengths, use the g. 9 Formula. 5. The segments joining the midpoints of onseutive sides of an isoseles trapezoid form a rhomus. a g. See margin. iven: Trapezoid T with T > ; D, E, F, and are midpoints of the indiated sides. rove: DEF is a rhomus. lan: The trapezoid is a. 9, so plae one ase on the. 9 so that the. 9 (-, ) D T(-a, 0) E (, ) F (a, 0) isets its ases. Use multiples of to name oordinates sine d. 9 will e involved. rhomus is a parallelogram with four e. 9. To show opposite sides are parallel, show that their f. 9 are the same. To show sides are ongruent, use g. 9. Developing roof Follow the plans aove to omplete the oordinate proofs. 6. (Eerise 3) The diagonals of an isoseles trapezoid are ongruent. roof: B the Distane Formula, E = a. 9 and HF =. 9. Therefore, E > HF the definition of ongruene. a. See margin. 7. (Eerise 4) The median from the verte of the right angle of a right triangle is half as long as the hpotenuse. roof: B the Distane Formula, = a. 9 and =. 9. Therefore, =. a. See margin. 350 Chapter 6 Quadrilaterals Quik Chek. a. (, ), (a ± d, ); starting with multiples of ou eliminate frations when using the midpoint formula , 0, 0; the are.. d ± a, T a, d ; so ± T d ± a whih is twie. So the midsegment is half the sum of the lengths of the ases. d. The ase along the -ais allows us to alulate length sutrating values. 5. a. isos.. -ais. -ais d. midpoints e. sides f. slopes g. the Distane Formula

4 8. (Eerise 5) The segments joining the midpoints of onseutive sides of an isoseles trapezoid form a rhomus. a k. See margin. roof: The midpoints have oordinates a. D(9, 9), E( 9, 9), F( 9, 9), and ( 9, 9). B the Distane Formula, DE =. 9, EF =. 9, F = d. 9, and D = e. 9. The slope of DE = f. 9 and the slope of F = g. 9. The slope of EF = h. 9 and that of D = i. 9. Thus, DEF is a parallelogram with ongruent j. 9, so k. 9 is a rhomus the definition of rhomus. 9a.(a, ) 9. Developing roof Use oordinate geometr.(a, ) to prove that the diagonals of a B(0, ).the same point retangle iset eah other. C(a, ) roof: The midpoint of C is a. 9. The midpoint of BD is. 9. The midpoints are. 9, so the diagonals iset eah other. D(a, 0) B ppl Your Skills 0. pen-ended ive an eample of a statement that ou think is easier to prove with a oordinate geometr proof than with a paragraph, flow, or two-olumn proof. Eplain our hoie. See ak of ook. roof. rove: The midpoints of the sides of a kite rolem Solving Hint S determine a retangle. E(0, a) ines with undefined K iven: Kite DEF with DE = EF and slope, like K and, D = F; K,,, and are D F(, ) are vertial lines. ll vertial lines are parallel. midpoints of the sides. rove: K is a retangle. See ak of ook. Visual earners Eerise Disuss with students wh point D must have oordinates (-, ). Use this disussion to help assess students understanding of oordinate geometr and how it illustrates what the learned aout kites in esson 6-5. Eerise 4 fter students answer the question, have them desrie a line for eah slope shown. Eerise 35 eview, if neessar, how to find the equation of a line. Eerise 36 Before students egin, ask: When two nonvertial lines are perpendiular, what is the produt of their slopes? Tatile earners Eerise 40 Have students eperiment with lassroom ojets, suh as pens, penils, and ooks, using trial and error to identif the entroids. State whether eah tpe of onlusion shown here ould e reahed using oordinate methods. ive a reason for eah answer. 3. See ak of ook.. B > CD 3. B 6 CD 4. B # CD 5. B isets CD. 6. B isets &CD. 7. & > &B 8. & is a right angle. 9. B + BC = C 0. #BC is isoseles.. & and &B are supplementar.. B, CD, and EF are onurrent. 3. Quadrilateral BCD is a rhomus. 4. ultiple Choie DEF is a rhomus. What is the slope of its diagonal DF? D 0 Q a D(0, a) E(, 0) undefined F nline Homework Help Visit: HShool.om We Code: aue a. "( a). "(a ) 7. a. "a. "a and B have oordinates and 0 on a numer line. Find the oordinates of the points that separate B into the given numer of ongruent segments n 5 9. See margin. 8. a. D( a, ), E(0, ), F(a ±, ), (0, 0). "(a ). "(a ) d. "(a ) esson 6-7 roofs Using Coordinate eometr 35 e. "(a ) f. a g. a h. a i. a j. sides k. DEF 5., 4, ,, 4, 6, , 0.4,.6,.8, 4, 5., 6.4, 7.6, ,.5,.8,..., 9.5, ± n, ± ( n ), ± 3( n ),..., ± (n )( n ) 35

5 4. ssess & eteah oweroint 33. (.76, 5.), (.5, 5.4), (.8, 5.6),..., (8.5, 4.6), (8.76, 4.8) ( 3 ± n, 5 ± n ), ( 3 ± ( ), 5 ± ( 0 n )),...,( 3 ± (n )( n n ), 0 5 ± (n )( n )) 35 esson Quiz Use the diagram for Eerises 5. (a, 0) B(, 0) C(, d ). oint is the midpoint of C. Find its oordinates. (a ±, d). oint is the midpoint of BC. Find its oordinates. ( ±, d) 3. Eplain how ou know that 6 B. Both have slope 0, so the are parallel. 4. Show that = B. The Distane Formula finds a and B a. 5. What theorem do Eerises 4 prove? Triangle idsegment Theorem lternative ssessment Have students work with partners to draw and identif the oordinates of a parallelogram in a oordinate plane. Then ask them to use oordinate geometr to prove an one theorem aout parallelograms from esson (0, 7.5), (3, 0), (6,.5) The endpoints of B are ( 3, 5) and B(9, 5). Find the oordinates of the points that separate B into the given numer of ongruent segments. 3. (, 63), (, 8 3 ), (3, 0), (5, n 3 ), (7, 33) See left See margin. 3. (.8, 6), ( 0.6, 7), roof (0.6, 8), (.8, 9), 35. You learned in esson 5-3 (Theorem 5-8) that the entroid of a triangle, (3, 0), (4., ), the point where the medians meet, is 3 of the distane from eah verte (5.4, ), (6.6, 3), to the midpoint of the opposite side. Complete the following steps (7.8, 4) to prove this theorem. a e. See margin. a. Find the oordinates of points,, and B(, d), the midpoints of the sides of #BC. * ) * ) * ). Find equations of, B, and C.. Find the oordinates * ) of point * ), the intersetion of and. * B ) d. Show that point is on C. e. Use the Distane Formula to show that C(, 0) point is 3 of the distane from eah verte to the midpoint of the opposite side. roof 36. Complete the following steps to prove rolem Solving Hint Theorem 5-9. You are given #BC with C(0, ) To show three lines are altitudes p, q, and r. Show that p, q, and r p onurrent, ou an r interset in a point (alled the orthoenter show of the triangle). a h. See ak of ook. ) one point is on all three lines a. The slope of BC is. (a, 0) B(, 0) What is the slope of line p? q ) the intersetion point of two lines is. Show that the equation of line p is = ( - a). on the third line, or. What is the equation of line q? 3) the intersetion of a one pair of lines is d. Show that lines p and q interset at Q 0,. the same as the e. The slope of C is. What is the slope of line r? intersetion of a a another pair. f. Show that the equation of line r is = ( - ). a g. Show that lines r and q interset at Q 0,. h. ive the oordinates of the orthoenter of #BC. eal-world C Challenge Connetion Carefull alaned metal shapes hang from wires in the moiles of leander Calder ( ). 35 Chapter 6 Quadrilaterals 35. a. (, d), ( +, d), (, 0) d. 4 : = ; d B 4 : = ( ); 4 C: = d ( ); The endpoints of B are as given. Find the oordinates of the points that separate B into n ongruent segments See ak of ook. 37. has oordinate a and B has oordinate on a numer line. 38. has oordinates (a, ) and B has oordinates (, d) in the oordinate plane. 39. Use the diagram at the right. a. Eplain using area wh ad = and hene ad =.(Hint: rea of triangle =? ase? height). Use slope and part (a) to show: If a =, then ad =. d a. See ak of ook. 40. hsis For a moile to e in alane, ou suspend eah part of the moile at its enter of mass. The enter of mass, or entroid, is the point around whih the weight of an ojet appears to e evenl distriuted. You learned a method for finding the entroid of a triangle in esson 5-3. ow use it to help ou find the entroid of a quadrilateral. (Hint: Where is the entroid of a segment?) See margin. ( ). Q 3, d 3 d. t. satisfies the eqs. for 4 4 and C. e. = "( ) d ; ( ) = Î Q Q d 3 5 d "Q 3 Q( ) d = 3 "( ) d = 3 The other distanes are found similarl. a

6 Test rep roof 4. Write a oordinate proof of the theorem: If the slopes of two lines have produt -, the lines are perpendiular. a. First, argue that neither line an e horizontal or vertial.. Then, tell wh the lines must interset. (Hint: Use indiret reasoning.). Knowing that the do interset, plae the lines in the oordinate plane, hoose a point on,, find a related point on,, and omplete the proof. a. See ak of ook. Test rep esoures For additional pratie with a variet of test item formats: Standardized Test rep, p. 36 Test-Taking Strategies, p. 356 Test-Taking Strategies with Transparenies ultiple Choie Short esponse Etended esponse ied eview for Help esson 6-6 esson 5-4 esson 4-4 lesson quiz, HShool.om, We Code: aua Divide the quad. into >. Find the entroid for eah k and onnet them. ow divide the quad. into other > and follow the same steps. Where the two lines meet onneting the entroids of the 4 > is the entroid of the quad. 4. Two points on a line are (-7, 0) and (9, ). Two points on a line parallel to that line are (, -3) and (, 4). What is the value of?. -3 B. 3 C. 5 D Two points on a line are (-4, 0) and (8, 8). Two points on a line perpendiular to that line are (8, -) and (6, ). What is the value of? F. 3. H. 7 3 J The endpoints of a segment are (7, -3) and (a, ). The midpoint is (3, 4). a. What are the oordinates of the other endpoint? Show our work.. What is the length of the segment? Show our work. See ak of ook. 45. iven: #BC;, Q, and are the midpoints of B, C, and BC, respetivel. a. lae #BC in the oordinate plane writing oordinates for, B, and C. a. See margin.. What are the oordinates of, Q, and?. Use oordinate geometr to prove #Q > #Q SSS. 46. etangle at the right is entered at the origin. ive oordinates for point without using an new variales. ( a, ) Write (a) the inverse and () the ontrapositive of eah statement See margin. 47. If the sum of the angles of a polgon is not 3608, then the polgon is not a quadrilateral. (a, -) 48. If = 5, then = If a = 5, then a = If, -4, then is negative. 5. If. 0, then is positive. Eplain how ou an use SSS, SS, S, or S with CCTC to prove eah statement true See margin. 5. B > CB 53. & > & 54. &K > & C E B H F D esson 6-7 roofs Using Coordinate eometr [4] a. Sample:. "( a) 5 Q B (, ) Q Q (, ) (a, ) Q a kq kq SSS. [3] minor omputational error Q (a, 0) [] parts a and orret C (0, 0) (a, 0) [] one part orret K 47. a. If the sum of the ' of a polgon is 360, then the polgon is a quad.. If a polgon is a quad., then the sum of its ' is a. If u 5, then u 0.. If u 0, then u a. If a u 5, then a u 5.. If a u 5, then a u a. If 4, then is not neg.. If is not neg., then a. If K 0, then is not pos.. If is not pos., then K l lc, D CD and ldb lcdb so S kdb kcdb and CCTC B CB. 53. HE F, EF H, and HF HF the efleive rop. of, so khef kfh SSS. Then CCTC l l. 54. K is given, the efleive rop., and lk l all rt. ' are. So kk k SS, and lk l CCTC. 353