306 !"# %&' ()*+,-.' /'+0''- 12)-+* AflnalappllcaLlonofLheyLhagorean1heoremlsonLhecoordlnaLeplane.WecaneasllyflndLhe dlsLancebeLweenLwopolnLsverLlcallyorhorlzonLallyonacoordlnaLeplane[usLbycounLlng,buLflndlngLhe exacL dlsLance dlagonally we have noL been able Lo do unLll now. !"# %&'()*+# ,#(-##* .*/ !-0 10&*(' CnacoordlnaLeplane,wecannowflndLhedlsLancebeLweenanyLwopolnLsbydrawlnglnarlghL Lrlangle and uslng Lhe yLhagorean 1heorem.Conslder Lhe followlng example: noLlceLhaLlfwewanLLoflndLhedlsLancebeLween LheseLwopolnLs, !!"!# and $%",weneedLoflndLhelengLhof '.AlsonoLeLhaL ( lsLhe horlzonLaldlsLancebeLweenLhepolnLsand ) lsLheverLlcaldlsLancebeLween Lhe polnLs.WlLh all Lhose values we now have a rlghL Lrlangle and can use Lhe yLhagorean 1heorem as follows: (* + )* , '* -* + .* , '* / + 0& , '* !% , '* % , ' So we know LhaL Lhe dlsLance beLween Lhese polnLs ls flve unlLs.Whlle Lhls ls easy Lo see when drawn ouL onLhecoordlnaLeplane,LhereareLlmeswhenweareglvenLheLwopolnLswlLhouLaplcLure.lnLhaLcase,we have Lwo opLlons.We can elLher draw Lhe polnLs on Lhe coordlnaLe plane as above, or we can flnd Lhe horlzonLal and verLlcal dlsLance beLween Lhe polnLs ln anoLher way. 1o do Lhls wlLhouL graphlng, we reallze LhaL Lhe horlzonLal dlsLance beLween Lwo polnLs ls Lhe dlfference lnLhelr 1 values.WhylsLhls?Slmllarly,LheverLlcaldlsLancebeLweenLwopolnLslsLhedlfferencelnLhelr 2 values.Agaln, can you explaln why? SoleL'slookaLourLwopolnLsagaln, $!"!# and $%".1hehorlzonLaldlsLancewouldbeLhedlfference beLween2and3.SlncedlfferencemeanssubLracL,wecanLake % 3 ! , - LoflndLhehorlzonLaldlsLancels3.SlmllarlywecansubLracLLhe 2 valuesLogeL & 3 ! , . meanlngaverLlcaldlsLanceof4.WecanLhenplugln3 and 4 lnLo Lhe yLhagorean 1heorem and solve exacLly as above. ' ( ) Lesson 3 Notes: Pythagorean Theorem inthe Coordinate PlaneHint:d: diagonal is the same as ch: horizontal is the same as a (or b)v: vertical is the same as b (or a)a+ b = c22 2SAME