hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity Theorem (.2) Tringle Proportionlity Theorem (.3) onverse of the Tringle Proportionlity Theorem (.3) Proportionl Segments Theorem (.3) Tringle Midsegment Theorem (.3) Side-ngle-Side Similrity Theorem (.2) ngle isetor/proportionl Side Theorem (.3) Right Tringle/ltitude Similrity Theorem (.) Right Tringle ltitude/ Hypotenuse Theorem (.) Right Tringle ltitude/leg Theorem (.).1 ompring the Pre-imge nd Imge of iltion diltion inreses or dereses the size of figure. The originl figure is the pre-imge, nd the dilted figure is the imge. pre-imge nd n imge re similr figures, whih mens they hve the sme shpe ut different sizes. diltion n e desried y drwing line segments from the enter of diltion through eh vertex on the pre-imge nd the orresponding vertex on the imge. The rtio of the length of the segment to vertex on the pre-imge nd the orresponding vertex on the imge is the sle ftor of the diltion. sle ftor greter thn 1 produes n imge tht is lrger thn the pre-imge. sle ftor less thn 1 produes n imge tht is smller thn the pre-imge. Y 5 3.5 Y9 5 7.7 Y enter of diltion Y 5 2.5 Y9 5? Y 5? Y9 5 3.3 Y 5 1.0 Y9 5? sle ftor 5 Y9 Y Y9 5 2.2 Y Y9 5 2.2 Y Y9 5 2.2 Y 5 7.7 3.5 Y9 5 2.2 Y Y9 5 2.2Y Y9 5 2.2Y 5 2.2 Y9 5 2.2(2.5) Y9 5 Y 2.2 Y9 5 2.2(1.0) Y9 5 5.5 3.3 5 Y 2.2 Y9 5 2.2 1.5 5 Y 325
.1 ilting Tringle on oordinte Grid The length of eh side of n imge is the length of the orresponding side of the pre-imge multiplied y the sle ftor. On oordinte plne, the oordintes of the verties of n imge n e found y multiplying the oordintes of the verties of the pre-imge y the sle ftor. If the enter of diltion is t the origin, point (x, y) is dilted to (kx, ky) y sle ftor of k. y The enter of diltion is the origin. 16 1 L9 The sle ftor is 2.5. J(6, 2) J9 (15, 5) 12 K(2, ) K9 (5, 10) 10 K9 L(, 6) L9 (10, 15) 8 6 L K J9 2 J 0 2 6 8 10 12 1 16 x.1 Using Geometri Theorems to Prove tht Tringles re Similr ll pirs of orresponding ngles re ongruent nd ll orresponding sides of similr tringles re proportionl. Geometri theorems n e used to prove tht tringles re similr. The lternte Interior ngle Theorem, the Vertil ngle Theorem, nd the Tringle Sum Theorem re exmples of theorems tht might e used to prove similrity. y the lternte Interior ngle Theorem, / > / nd / > /. y the Vertil ngle Theorem, / > /. Sine the tringles hve three pir of orresponding ngles tht re ongruent, the tringles hve the sme shpe nd >. 326 hpter Similrity Through Trnsformtions
.1 Using Trnsformtions to Prove tht Tringles re Similr Tringles n lso e proven similr using sequene of trnsformtions. The trnsformtions might inlude rotting, dilting, nd refleting. Given: i F Trnslte so tht ligns with F. Rotte 180º out the point so tht gin ligns with F. Trnslte until point is t point F. If we dilte out point to tke point to point, then will e mpped onto, nd will e mpped onto F. Therefore, is similr to F. F.2 Using Tringle Similrity Theorems Two tringles re similr if they hve two ongruent ngles, if ll of their orresponding sides re proportionl, or if two of their orresponding sides re proportionl nd the inluded ngles re ongruent. n inluded ngle is n ngle formed y two onseutive sides of figure. The following theorems n e used to prove tht tringles re similr: The ngle-ngle () Similrity Theorem If two ngles of one tringle re ongruent to two ngles of nother tringle, then the tringles re similr. The Side-Side-Side (SSS) Similrity Theorem If the orresponding sides of two tringles re proportionl, then the tringles re similr. The Side-ngle-Side (SS) Similrity Theorem If two of the orresponding sides of two tringles re proportionl nd the inluded ngles re ongruent, then the tringles re similr. Given: / > / / > /F F Therefore, F y the Similrity Theorem. hpter Summry 327
.3 pplying the ngle isetor/proportionl Side Theorem When n interior ngle of tringle is iseted, you n oserve proportionl reltionships mong the sides of the tringles formed. You n pply the ngle isetor/proportionl Side Theorem to lulte side lengths of iseted tringles. ngle isetor/proportionl Side Theorem isetor of n ngle in tringle divides the opposite side into two segments whose lengths re in the sme rtio s the lengths of the sides djent to the ngle. The mp of n musement prk shows lotions of the vrious rides. Flying Swings Ride Given: Pth isets the ngle formed y Pth nd Pth. Pth rousel Pth Pth Tower rop Ride Pth is 13 feet long. Pth is 65 feet long. Pth Pth Pth is 55 feet long. Roller oster Let x equl the length of Pth. x 5 13 Pth is 121 feet long. 55 63 x 5 121.3 pplying the Tringle Proportionlity Theorem The Tringle Proportionlity Theorem is nother theorem you n pply to lulte side lengths of tringles. Tringle Proportionlity Theorem If line prllel to one side of tringle intersets the other two sides, then it divides the two sides proportionlly. Given: H i G 5 GH F FG 5 30 FG 5 GH? F F 5 5 5 (25)(5) 30 GH 5 25 5 37.5 FG 5? F H G 328 hpter Similrity Through Trnsformtions
.3 pplying the onverse of the Tringle Proportionlity Theorem The onverse of the Tringle Proportionlity Theorem llows you to test whether two line segments re prllel. onverse of the Tringle Proportionlity Theorem If line divides two sides of tringle proportionlly, then it is prllel to the third side. F G Given: 5 33 5 GH H F FG F 5 11 33 5 66 11 22 GH 5 22 3 5 3 FG 5 66 Is H i G? pplying the onverse of the Tringle Proportionlity, we n onlude tht H i G..3 pplying the Proportionl Segments Theorem The Proportionl Segments Theorem provides wy to lulte distnes long three prllel lines, even though they my not e relted to tringles. Proportionl Segments Theorem If three prllel lines interset two trnsversls, then they divide the trnsversls proportionlly. Given: L 1 i L 2 i L 3 5 52 5 26 L 1 5 0 F 5? 5 F? F 5? F L 2 L 3 F 5? 5 (0)(26) 52 5 20 hpter Summry 329
.3 pplying the Tringle Midsegment Theorem The Tringle Midsegment Theorem reltes the lengths of the sides of tringle when segment is drwn prllel to one side. Tringle Midsegment Theorem The midsegment of tringle is prllel to the third side of the tringle nd hlf the mesure of the third side of the tringle. F Given: 5 9 F 5 9 FG 5 11 GH 5 11 H 5 17 G Sine 5 F nd FG 5 GH, point is the midpoint of F, nd G is the midpoint of FG. G is the midsegment of F. H G 5 1 2 H 5 1 (17) 5 8.5 2. Using the Geometri Men nd Right Tringle/ltitude Theorems Similr tringles n e formed y drwing n ltitude to the hypotenuse of right tringle. Right Tringle/ltitude Similrity Theorem If n ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. The ltitude is the geometri men of the tringle s ses. The geometri men of two positive numers nd is the positive numer x suh s x 5 x. Two theorems re ssoited with the ltitude to the hypotenuse s geometri men. The Right Tringle ltitude/hypotenuse Theorem The mesure of the ltitude drwn from the vertex of the right ngle of right tringle to its hypotenuse is the geometri men etween the mesures of the two segments of the hypotenuse. The Right Tringle ltitude/leg Theorem If the ltitude is drwn to the hypotenuse of right tringle, eh leg of the right tringle is the geometri men of the hypotenuse nd the segment of the hypotenuse djent to the leg. x 7 15 7 x 5 x 15 x 2 5 105 x 5 105 10.2 330 hpter Similrity Through Trnsformtions
.5 Proving the Pythgoren Theorem Using Similr Tringles The Pythgoren Theorem reltes the squres of the sides of right tringle: 2 1 2 5 2, where nd re the ses of the tringle nd is the hypotenuse. The Right Tringle/ ltitude Similrity Theorem n e used to prove the Pythgoren Theorem. Given: Tringle with right ngle. onstrut ltitude to hypotenuse, s shown. ording to the Right Tringle/ltitude Similrity Theorem,. Sine the tringles re similr, 5 nd 5. Solve for the squres: 2 5 3 nd 2 5 3. dd the squres: 2 1 2 5 3 1 3 Ftor: 2 1 2 5 ( 1 ) Sustitute: 2 1 2 5 () 5 2 This proves the Pythgoren Theorem: 2 1 2 5 2.5 Proving the Pythgoren Theorem Using lgeri Resoning lgeri resoning n lso e used to prove the Pythgoren Theorem. Write nd expnd the re of the lrger squre: ( 1 ) 2 5 2 1 2 1 2 Write the totl re of the four right tringles: ( 1 2 ) 5 2 Write the re of the smller squre: 2 Write nd simplify n eqution relting the re of the lrger squre to the sum of the res of the four right tringles nd the re of the smller squre: 2 1 2 1 2 5 2 1 2 2 1 2 5 2 hpter Summry 331
.5 Proving the onverse of the Pythgoren Theorem lgeri resoning n lso e used to prove the onverse of the Pythgoren Theorem: If 2 1 2 5 2, then nd re the lengths of the legs of right tringle nd is the length of the hypotenuse. Given: Tringle with right ngle. Relte ngles 1, 2, 3: m/1 1 m/2 1 m/3 5 180º Use the Tringle Sum Theorem to determine m/1 1 m/2. m/1 1 m/2 5 90º etermine m/3 from the smll right ngles: Sine m/1 1 m/2 5 90º, m/3 must lso equl 90º. 2 3 1 1 3 2 2 3 1 1 3 2 Identify the shpe of the qudrilterl inside the lrge squre: Sine the qudrilterl hs four ongruent sides nd four right ngles, it must e squre. etermine the re of eh right tringle: 5 1 2 etermine the re of the enter squre: 2 Write the sum of the res of the four right tringles nd the enter squre: ( 1 2 ) 1 2 Write nd expnd n expression for the re of the lrger squre: ( 1 ) 2 5 2 1 2 1 2 Write nd simplify n eqution relting the re of the lrger squre to the sum of the res of the four right tringles nd the re of the smller squre: 2 1 2 1 2 5 2 1 2 2 1 2 5 2.6 Use Similr Tringles to lulte Indiret Mesurements Indiret mesurement is method of using proportions to lulte mesurements tht re diffiult or impossile to mke diretly. knowledge of similr tringles n e useful in these types of prolems. Let x e the height of the tll tree. x 5 32 20 18 x 5 (32)(20) 18 x 35.6 20 feet 50 50 The tll tree is out 35.6 feet tll. 18 feet 32 feet 332 hpter Similrity Through Trnsformtions