OVERVIEW Using Similarity and Proving Triangle Theorems G.SRT.4

Transcription

1 OVRVIW Using Similrity nd Prving Tringle Therems G.SRT.4 G.SRT.4 Prve therems ut tringles. Therems include: line prllel t ne side f tringle divides the ther tw prprtinlly, nd cnversely; the Pythgren Therem prved using tringle similrity. This jective intrduces wht is smetimes knwn s the side splitting therem r prprtinl prts therem. Lk t the impct tht prllel lines hve n dividing the lines up prprtinlly. Prprtinl vlues cn e fund in ther plces in the tringle when prllel lines re frmed. (1) The student will prve (the side splitting therem) tht line prllel t ne side f tringle divides the ther tw prprtinlly. () The student will prve (the ngle isectr therem) tht n ngle isectr f n ngle f tringle divides the ppsite side in tw segments tht re prprtinl t the ther tw sides f the tringle. (3) The student will prve the Pythgren Therem using similrity nd the gemetric men. Fcus n student utcme 1 mre thn. The ngle isectr hs very limited use utside f its wn sic prlems. mphsize the side splitting therem nd its use in multiple settings. 1 The mst cmmn errr in this unit is fund in this jective Students ften frm scle fctr etween tw prprtinl pieces nd then relte it t tw prprtinl sides. These tw scle fctrs re nt equl nd this cuses mny errrs. The ngle isectr therem while quite pretty t prve nd quite esy t use hs very little pplictin eynd itself. The min fcus here is the side splitting therem. gsrt4usingsimilrityndprvingtringletherems Pge 1 f 7 4/9/014

2 NOTS Using Similrity nd Prving Tringle Therems G.SRT.4 ONPT 1 Prve therems ut tringles The Side Splitting Therem - line prllel t ne side f tringle divides the ther tw prprtinlly. This reltinship is smetimes clled the Side Splitting therem. Given: Prve: Given tht m m y rrespnding m m y rrespnding Thus y. Similrity rings prprtinl sides. press the distnce s the sum f its tw pieces, = + (using segment dditin) nd is the sum f its tw pieces, = + (using segment dditin) S y sustitutin, it fllws tht: ( + ) = ( + ) y crss multiplictin () + () = () + () y distriutin Sutrct () frm th sides leving () = () gsrt4usingsimilrityndprvingtringletherems Pge f 7 4/9/014

3 NOTS Using Similrity nd Prving Tringle Therems G.SRT.4 ONPT Prve therems ut tringles The ngle isectr Therem n ngle isectr f n ngle f tringle divides the ppsite side in tw segments tht re prprtinl t the ther tw sides f the tringle. Given: where is n ngle isectr f. Prve: rete n uiliry prllel line t thrugh Pint while ls etending side until the tw meet t pint. m m ecuse rrespnding ngles, nd m m ecuse lternte Interir ngles = ecuse f the Issceles Tringle Therem Thus using the side splitting therem nd then using sustitutin. gsrt4usingsimilrityndprvingtringletherems Pge 3 f 7 4/9/014

4 NOTS Using Similrity nd Prving Tringle Therems G.SRT.4 ONPT 3 Prve therems ut tringles. Prve the Pythgren Therem using tringle similrity. Given: right tringle with n ltitude (height) drw frm the right ngle t the hyptenuse. Prve: c d c e When cmpring the left inner tringle t the entire tringle, ntice tht th tringles hve cmmn ngle nd th hve right ngle. Thus they re similr y. Knwing tht they re similr llws fr the estlishment f the prprtin etween the sides. d c left d cd whle c When cmpring the right inner tringle t the entire tringle, ntice tht th tringles hve cmmn ngle nd th hve right ngle. Thus they re similr y. Knwing tht they re similr llws fr the estlishment f the prprtin etween the sides. c e right e whle c Nw the pretty prt!!! It is nw knwn tht nd tht ce cd ce The Pythgren Therem is eginning t pper. Net, y dding th equtins tgether. Very clse t the Pythgren Therem Just fctr ut c. c d e ( ) frm the digrm ntice tht length c = d + e nd sustitute tht in.. WOW!!! c gsrt4usingsimilrityndprvingtringletherems Pge 4 f 7 4/9/014

5 SSSSMNT Using Similrity nd Prving Tringle Therems G.SRT.4 1. In the given digrm, which f the fllwing sttement is NOT true: ) ) ) ). In the given digrm, which f the fllwing sttement is NOT true: G ) ) ) F G G ) F G G F F 3. Which f the fllwing wuld nt slve fr the crrect vlue f : ) 5 5 ) ) 5 5 ) is the sme prprtin s T r F 5. Tim clims tht ecuse, tht the prprtin Jennifer disgrees with Tim. Wh is crrect? Why? If Jennifer is crrect, rewrite the prprtin s tht it is true. is vlid. 6. mplete the prprtins. ) ) c) G FG d) e) f) FG G F G F G gsrt4usingsimilrityndprvingtringletherems Pge 5 f 7 4/9/014

6 SSSSMNT Using Similrity nd Prving Tringle Therems G.SRT.4 7. Find the vlues fr the missing vriles. ) ) c) d) = = = =. etermine the vlues fr the vriles, nd y y gsrt4usingsimilrityndprvingtringletherems Pge 6 f 7 4/9/014

7 SSSSMNT Using Similrity nd Prving Tringle Therems G.SRT.4 nswers: 1) ) 3) 4) T 5) Jennifer. The tw similr tringles re, which mkes the crrect prprtin. The errr tht Tim mde ws tht the mesurements were ll full lengths f sides ecept, it is nly ( full) ( full) prtin f the side thus is n incrrect prprtin. ( prt) ( full) 6) ) ) F c) d) e) f) G 7) ) = 6.4 ) = 11.5 c) = 4 d) = 16 4 ) 10 4 s = 9.6. Use the Pythgren Therem t slve fr y. Finlly t slve fr y using, 10 6 results in y = y 10 4 m therefre m = 6. gsrt4usingsimilrityndprvingtringletherems Pge 7 f 7 4/9/014