F gur 4-22 Figuares halint~g area.
is fsby nog6 meanrS as top ae exerctise ine geom~et8gry to Prov`ke that thPe r~esu t doss
not depend on usat w6 weCut up~f te fiCgure.~
Fairrom thkese; exs;ama-ples, wvje miiightb~ h) led to expe~ct that area can hpe de ned
or ererU f~igureb in the plane. If by "everyg figurpe"" wie nwan every~ set of
pointsin; the pl~ane, t-hen we havae gonle too~ far. ForP ehe poaints-r nk the frt
quad~rant, in terms of xU-cordirpnates, ~Fonus a st% to wh14~ Wh w1e. canhad
assign a finite area Other* exa 40es @%an he ~iven of strange bouazndd set
of po~intas for wich th~e a2ssignmnent of ~area nsta reason~ahe 8%Iav is in doubt.
Mclodern resear~ches have sh-ows~n th-at thke best we ca at tain is certl~ai
c ass of figre Shavmg$b area.t Furhermore, for th~ese iguresa~, we hav~e ~thef
owia~tng g~reslt$s (whih themse v~~%es a71~ ow us o say w have do nod~b areak:
Ru eL f, on~ e figure is co0ngrupent tB aother and th~e fi~ars Agure~- hs are,
then th-ke seco~nd figurae hs area an-d th~e twpo arleas ar~e equaL
u e.I fg two0 figures5 have area ram the fir~stfure achue~ds te: secndJ
thena th-e area of the~ first- figure iS at leat ~as large a that of thesecnd:
further more, zthe figur~e obta ned byr remdxoving~ the seeonsd pgure from theza
first has area, equgal~~ to th~8at of the first. igure m asilS that oDf the SeCo
Ruj% e. If a givaena figu~re is fonured of a fin to nunaborde o~f i~gurs, no- two
of$ wx~ih havJe poaints in corm;ninn, aind each of: whd ch6 haaea, thaen- th
~given re~k6 has area, equa to3 the: Surn~i of ~he sareas of ht~e seorra fires o~f
wh=I t is compo~sed.
RueI. Th~e arcea o a rectanTe eof des a, b.s T~,e area f se e pty
RrBe V. If a fg~iven Egure s inclu~ded in other aru~et having area anda% if
th~e ~venB fgurse ine udiles oth~aer figures hradg ~are an~d fS te greatest o~we
bound of the areas o~f the indcudci~ng fgUrPE~S equs 6a e least upp-er how of
thge area of the inouded fi~gulre~s, hen~ the given agre haas aea, equa tb he
cornunaonB va neF oBBf t e6f gr atest ower~ boud am e least uppr~er boud.
In hd~ 1 rthe woard ""conbgruentk" is used as in geonwPstry.g~ Xare we~i
hgurs are csnagruene if, and only it t~here fso <w oa efoone eorrsspondence
betwesen the~ f gres so thataa each~& pair of points of the pus ~iare eof~crEs,~ZPonds.~
to an~ equiditstantr pair9 of poinrts od the seconcd gu~re. I~e pweseur ao the
oth-er ru es, the fi~rst par~t: of Ru me IV is equiva cu~et to gikng %he usad n ~e:
for th~-e aresa of a tr and~le. Rule V6 is su gested8 by t-h jariars proce" o
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