Why? DF = 1_ EF = _ AC

Transcription

1 Similr Tringles Then You solved proportions. (Lesson 2-) Now 1Determine whether two tringles re similr. 2Find the unknown mesures of sides of two similr tringles. Why? Simon needs to mesure the height of Ferris wheel for clss project. Simon cn mesure her shdow nd the shdow of the Ferris wheel. She cn then use similr tringles nd indirect mesurement to find the height of the Ferris wheel. New Vocbulry similr tringles Similr Tringles Similr tringles hve the sme shpe, but not necessrily 1 the sme size. The symbol is used to denote tht two tringles re similr. The vertices of similr tringles re written in order to show the corresponding prts. Key Concept Similr Tringles Virgini i SOL Preprtion for G. The student, given informtion in the form of figure or sttement, will prove two tringles re similr using lgebric nd coordinte methods s well s deductive proofs. Words Emple If two tringles re similr, then the mesures of their corresponding ngles re equl, nd the mesures of their corresponding sides re proportionl. If ABC DEF, then m A = m D, m B = m E, m C = m F, nd AB_ DE = _ BC EF = _ AC DF = 1_ You cn determine similrity of two tringles by nlyzing the corresponding ngles. If the corresponding ngles hve the sme mesure, then the tringles re similr. Emple 1 Determine Whether Two Tringles re Similr Determine whether the pir of tringles is similr. Justify your nswer. The mesure of T is ( + ) or. In XYZ, X nd Z hve the sme mesure. Let = the mesure of X nd Z. + + = 180 Sum of the ngle mesures is = 11 Subtrct from ech side. = Divide ech side by 2. Since, m X = nd m Z =, XYZ STQ. GuidedPrctice 1. Determine whether ABC with m A = 8 nd m B = m C is similr to DEF with m E = m F =. Justify your nswer. 2 Lesson 10-

2 The rtios of the lengths of the corresponding sides cn lso be compred to show tht two tringles re similr. Emple 2 Determine Whether Two Tringles re Similr Determine whether the pir of tringles is similr. Justify your nswer. If VXZ nd WXY re similr, then the mesures of their corresponding sides re proportionl. _ VX WX = _ = _ XZ XY = _ 1 = _ VZ WY = _ 9 = 9 V W Y 1 X Since the corresponding sides re proportionl, VXZ WXY. Z Mth History Link Pythgors (80 00 B.C.) Greek mthemticin Pythgors founded the fmous Pythgoren school for study of philosophy, mthemtics, nd nturl science. GuidedPrctice 2. Determine whether ABC with AB =, BC = 1, nd AC = 20 is similr to JKL with JK =, KL = 8, nd JL = 9. Justify your nswer. Find Unknown Mesures When some of the mesurements of the sides of 2 similr tringles re known, proportions cn be used to find the missing mesures. Emple Find Missing Mesures Find the missing mesures for the pir of similr tringles. 8 c _ AB PQ = _ AC PR c 8 = Corresponding sides of similr tringles re proportionl. AB = c, AC =., PQ = 8, PR = 8 8 BC_ QR = _ AC PR = Corresponding sides of similr tringles re proportionl. BC =, AC =., QR =, PR = c = 100 Find the cross products. = 8. Find the cross products. c = 20 Divide ech side by. = 1. Divide ech side by. The missing mesures re 20 nd 1.. Problem-Solving Tip Drw Digrm When problem involves sptil resoning, or geometric figures, drw digrm. For emple, in Guided Prctice B, drw ech tringle seprtely to help determine the nswer. GuidedPrctice A. B. b connected.mcgrw-hill.com

3 Rel-World Creer Cruise Director The cruise director is in chrge of ll onbord entertinment nd ctivities. A professionl entertinment bckground is preferred or 2 yers eperience on bord. Rel-World Emple Indirect Mesurement SHADOWS Tori is feet inches tll, nd her shdow is 2 feet 9 inches long. She is stnding net to flgpole. If the length of the shdow of the flgpole is feet long, how tll is the flgpole? Understnd Find the height of the flgpole. Pln Mke sketch of the sitution. Solve The Sun s rys form similr tringles. Write proportion tht compres the heights of the objects nd the lengths of their shdows. Check Let = the height of the flgpole. height of flgpole _. = _ Tori s height 2. _ 2 _. 2.. =. 2. = = 2 Substitute 2 for. 2. ft. ft ft length of the flgpole s shdow Tori s shdow ft The height of the flgpole is 2 feet. GuidedPrctice. TENTS The directions for pitching tent include scle drwing in which 1 inch represents. feet. In the drwing, the tent is 1 _ inches tll. How tll should the ctul tent be? Check Your Understnding = Step-by-Step Solutions begin on pge R. Emples 1 2Determine whether ech pir of tringles is similr. Justify your nswer Emple Find the missing mesures for the pir of similr tringles if ABC XYZ. =, b =, c = 8, =. = 9, y = 1, z = 21, c = c z. = 2, b =, = 10, z = 0 b y. b =, c = 10, = 0, y = 1 Emple. TREES Mrl wnts to know the height of the tree in her bckyrd. The tree csts shdow 8 feet inches long. Mrl is feet tll, nd her shdow is 2 feet inches long. How tll is the tree? Lesson 10- Similr Tringles

4 Prctice nd Problem Solving Etr Prctice begins on pge 81. Emples 1 2Determine whether ech pir of tringles is similr. Justify your nswer Emple Find the missing mesures for the pir of similr tringles if HKM PTR. 1. m = 2, k =, h =, r = 1. r =., p = 1, t = 20, h = m r 1. m =., k = 9, t = 1., p = m = 1., h = 2.8, p = 0., t = m =, h = 2 2, t =, r = 21 k h t p 19. m = 2, k =, t = 1, p = 10 Emple B 20. TOYS Diecst model crs use scle of 1 inch : 2 feet of the rel vehicle. The originl vehicle hs window shped like right tringle. If the height of the window on the ctul vehicle is 2. feet, wht will the height of the window be on the model? 21. GOLF Betriz is plying miniture golf on hole like the one shown t the right. She wnts to putt her bll U so tht it will bnk t T nd trvel into the hole t R. Use similr tringles to find where Betriz s bll should strike the wll. 22. MAPS The scle on n Ohio mp shows tht 2. centimeters represents 100 miles. The distnce on the mp from Clevelnd to Cincinnti is. centimeters. About how mny miles prt re the two cities? in. R 0 in. U in. 2 in. S T 28 V W 2 SCHOOL PROJECT For etr credit in his history clss, Mrquez plns to mke model of the Sttue of Liberty in the scle 1 inch : 10 feet. If the height of the ctul Sttue of Liberty is 11 feet, wht will be the height of the model? 2. TENNIS Andy wnts to hit the bll just over the net so it will lnd 1 feet wy from the bse of the net. If Andy hits the bll 0 feet wy from the net, how high does he hve to hit the bll? net ft 1 ft 0 ft connected.mcgrw-hill.com

5 C 2 MULTIPLE REPRESENTATIONS In this problem, you will compre the rtios of corresponding sides nd perimeters of similr tringles.. Algebric Wht is the rtio of the corresponding sides of ech pir of similr tringles? Record your results in the tble. 2 8 b. Tbulr Find the perimeter of ech tringle. Then find the rtio of the perimeters for ech pir of tringles. Record your results in the tble. Pir 1 Pir 2 Pir 9 Similr Tringles Rtios of Sides Perimeters Rtios of Perimeters smller tringle lrger tringle smller tringle lrger tringle smller tringle lrger tringle c. Anlyticl How is the rtio of the perimeters relted to the rtio of the lengths of corresponding sides for ech pir of tringles? d. Anlyticl If the rtio of the lengths of corresponding sides of two similr tringles is 1:, wht would be the rtio of their perimeters? H.O.T. Problems Use Higher-Order Thinking Skills 2. ERROR ANALYSIS Kwm nd Roslind re compring the similr tringles. Is either of them correct? Eplin. Kwm m X = m T m Y = m U m Z = m V XYZ, TUV Roslind m X = m U m Y = m V m Z = m T XYZ, UVT 2. CHALLENGE Tringle XYZ is similr to the two tringles formed by the line segment from Z perpendiculr to XY, nd these two tringles re similr to ech other. Write three similrity sttements bout these tringles. Why re the tringles similr to ech other? 28. REASONING Is the sttement sometimes, lwys, or never true? Eplin. Lesson 10- Similr Tringles If the mesures of the sides of tringle re multiplied by, then the mesures of the ngles of the enlrged tringle will hve the sme mesures s the ngles of the originl tringle. 29. OPEN ENDED Drw nd lbel tringle ABC. Then drw nd lbel similr tringle PQR so tht the re of PQR is four times the re of ABC. Eplin your strtegy. 0. WRITING IN MATH Summrize how to determine whether two tringles re similr to ech other nd how to find missing mesures of similr tringles.

6 Virgini SOL Prctice A..b, A..d, G.., G. 1. Find the distnce between the points t (2, -) nd (-, 8). A B C 9 D GEOMETRY Find the vlue of if the two tringles re similr.. Which eqution represents line with y-intercept of - nd slope of? A y = - C y = - - B y = - + D y = +. SHORT RESPONSE Wht re the - nd y-intercepts of the function grphed below? y 1 F 2. G 10 H 19 J 21 Spirl Review Find the distnce between the points with the given coordintes. (Lesson 10-). (0, ), (1, 9). (-2, ), (, 1). (1, -), (-1, -) 8. (, -2), (-2, ) 9. (-, -), (-1, 2) 0. (-, -), (-, -8) Determine whether the mesures cn be the lengths of the sides of right tringle. (Lesson 10-) 1.,, 2. 8, 10,. 10, 2, 2.,, 1., 9, 1.,,. NUTRITION In the function y = , y represents the consumption of bred nd cerel in pounds per person in the United Sttes, nd represents the number of yers since If this trend continues, in wht future yer will the verge Americn consume 00 pounds of bred nd cerel? (Lesson 9-) Fctor ech polynomil, if possible. If the polynomil cnnot be fctored, write prime. (Lesson 8-) 8. k b g 2 + 0g t + t 2-8t. 20 n 2 + n +. DRIVING Averge speed is clculted by dividing distnce by time. If the speed limit on n interstte is miles per hour, how fr cn person trvel leglly in 1 1_ hours? (Lesson -2) 2 Skills Review Evlute if =, b = -2, nd c =. (Lesson 1-2). _ b 2b c. _ c. _ c -b 8. _ -c 2b 9. _ -2bc connected.mcgrw-hill.com