% %ȉ̕RgƂ̂łC_ł͂܂ł܂B
% ̃^Cg
[Title]
402_~Ɛ`

% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
% tHg̑傫B1`10C܂TeX̃R}hw肷B
% ftHǵC5i\normalsizej
% 1\tinyC \scriptsizeC3\footnotesizeC4\smallC 5\normalsize
% 6\largeC7\LargeC     8\LARGEC       9\hugeC 10\Huge
[FontSize]
5

% vAuɒǉpbP[Wt@Cw肷B
[usepackage]
\usepackage{color}

% ꂼ̖𓚂$\displaystyle $tꍇ́C@ON ܂1
% ꂼ̖𓚂$\displaystyle $tȂꍇ́COFF܂0
% up̕ҏWv|u[U[ݒv̉ɂݒ
% @@@@@@@@@@@@@@@@@ꍇɎw肵ĂB
% LqȂ΁Cup̕ҏWv|u[U[ݒv
% @@@@@@@@@@@@@@@@@@@ɂݒ肪D悳܂B
[displaystyle]
OFF


% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
{
̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic98.tex}
\end{center}

78

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic99.tex}
\end{center}

62

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic100.tex}
\end{center}

100

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic101.tex}
\end{center}

process
ڌ藝ɂC$\angle x=40$

40

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic102.tex}
\end{center}

188

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic103.tex}
\end{center}

58

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic104.tex}
\end{center}

85

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic105.tex}
\end{center}

process
ڌ藝ɂC$\angle x=46$

46

}́$x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic91.tex}
\end{center}

150

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic92.tex}
\end{center}

32

}́$x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic93.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic93p.tex}
\end{center}
$96+x=110$B

14

}́$x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic94.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic94p.tex}
\end{center}
$28+50=78$B

78

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic95.tex}
\end{center}

94

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic96.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic96p.tex}
\end{center}
$72+x=90$B

18

}́$x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic97.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic97p.tex}
\end{center}

58

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic01.tex}
\end{center}

process
ڌ藝ACB=45B\\
$60+45+x=180$

75

}́$x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic02.tex}
\end{center}

process
$\angle $ ACB=~p=Sp $ \div 2=50$ \\
AB=BEC$ \angle $ ACB $ = \angle$  BDE

50

̐}$\angle x$C$\angle y$߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic03.tex}
\end{center}

process
$y=85$B$z+85=180 $C$z=95$B\\
$x=180-95=85$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic03k.tex}
\end{center}

$x=85Cy=85$

}PQ͉~̐ڐłB$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic04.tex}
\end{center} 

$108$

}ɂāC_O͉~̒SłCPCQCR͂ꂼ~̓_łB̂ƂC$\angle x$̑傫߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic05.tex}
\end{center}

process
̌PRɑ΂钆Sp\\
$360-120=240$B
$x=240 \div 2=120$

$120$

}ɂāCaCb̊px߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic06.tex}
\end{center}

process
$\angle $OAC $=18$B$\angle $ ACO $=90$Ȃ̂ŁC$\angle $ COA $72$BāC$\angle $ COB@$=144$B

a$=108$Cb$=216$

}ŁCS_ACBCCCD~Ỏ~ɂC$\angle $ AOC $=130$̂ƂC$\angle x$C$\angle y$߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic07.tex}
\end{center}

process
ADCɑ΂钆Sp$\angle$ AOC $=130$@C
~p$\angle x=65$B$y+65=180$C$y=115$

$x=65Cy=115$

}ŁCS_ACBCCCD~Ỏ~ɂBAC~O̒aŁC$\angle$ ACB $=55$̂ƂC$\angle x$̑傫߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic08.tex}
\end{center}

process
$\angle CAB =90-55=35$B$\angle x=35$

$\angle x=35$

}ŁCS_ACBCCCD͉~Ỏ~ɂBST͉~O̐ڐłB$\angle$ CAD $=40$̂ƂC$\angle x$̑傫߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic09.tex}
\end{center}

process
~p̒藝C$\angle$ DBC $=40$B\\
ڌ藝C$\angle$ BDC$=x$BāC\\
$x+40+90=180$

$\angle x=50$

}$\angle b$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic10.tex}
\end{center}

process
~p̒藝C$\angle$ ABD $=50$B\\
$b=30+50$

$\angle b=80$

}$\angle d$C$\angle e$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic11.tex}
\end{center}

process
$d=180-95=85Ce=110$

$d=85Ce=110$

}ŁC$\angle x$̒l߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic13.tex}
\end{center}

$40$

}ŁC$\angle x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic14.tex}
\end{center}

$27$

}ŁC$\angle x$̒l߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic15.tex}
\end{center}

$140$

}ŁC$\angle x$̒l߂BC_AC_B͉~O̐ړ_łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic16.tex}
\end{center}

$50$

}ŁC$\angle x$̒l߂BCAC//OBłCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic17.tex}
\end{center}

$96$

}ŁC$\angle x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic18.tex}
\end{center}

process
$x+(x+20)=76$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic18p.tex}
\end{center}

$28$

}́CAB𒼌aƂ锼~OłB$\angle$ ACD $=15$C$\angle$ COB$=70$̂ƂC$\angle$ CAD̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic19.tex}
\end{center}

process
}C$x+35+90+15=180$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic19p.tex}
\end{center}

$40$

}ŁCCPCŉ~ɐڂĂB$\angle$ BPC $=25$C$\angle$ BAC $=30$̂ƂC$\angle$ ACB̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic20.tex}
\end{center}

process
ڌ藝C$\angle$ BCP = $\angle$ BAC $=30$BOp`̓p̘a$180$Ȃ̂ŁC
$(x+30)+30+25=180$ 

$95$

}̎lp`ABCD́C~ɓڂĂB̂ƂC$\angle$ CAD̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic21.tex}
\end{center}

process
$\angle $CDA $=180-73=107$B$x+41+107=180$

$32$

}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic22.tex}
\end{center}

process
$\angle $ BED $=95$B$x+15=95$

$80$

}ɂāC_O͉~̒SłCPQRS͂ꂼ~̓_łB
$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic26.tex}
\end{center}

process
$\angle$ QPS $=190\div 2=95$B\\
$\angle$ QPS $+$ QRS $=180$

$95$

}ST͉~O̐ڐłB$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic27.tex}
\end{center}

process
Op̒藝C$\angle$ PQS $+49=75$B\\
$\angle$ PQS $=26$B
ڌ藝C\\
$\angle$ PRQ $=26$

79

}ɂāC_O͉~̒SłCACBCC͂ꂼ~̓_łB
̂ƂC$\angle x$̑傫͂炩BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic131.tex}
\end{center}

$38$

}ɂāC_O͉~̒SłCACBCCCD͂ꂼ~̓_łB
̂ƂC$\angle x$̑傫͂炩BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic132.tex}
\end{center}

$40$

}ɂāClp`ABCD͉~ɓڂĂB
̂ƂC$\angle x$̑傫͂炩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic133.tex}
\end{center}

$102$

}ɂāC_O͉~̒SłCACBCCCD͂ꂼ~̓_łB
̂ƂC$\angle x$̑傫͂炩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic134.tex}
\end{center}

24

}ɂāCPACPC͐ڐłB
$\angle x$̑傫͂炩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic135.tex}
\end{center}

$66$

}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic141.tex}
\end{center}

$60$

}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic142.tex}
\end{center}

$20$

}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic143.tex}
\end{center}

$110$

}$\angle x$C$\angle y$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic144.tex}
\end{center}

$x=110$C$y=92$

}$\angle x$C$\angle y$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic145.tex}
\end{center}

$x=103$C$y=77$

}$\angle x$C$\angle y$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic146.tex}
\end{center}

$x=73$C$y=80$

}$\angle x$C$\angle y$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic147.tex}
\end{center}

$x=35$C$y=55$

}$\angle x$C$\angle y$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic148.tex}
\end{center}

$x=62$C$y=40$

}$\angle x$̑傫߂B
ȂC_O͉~̒SłBCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic149.tex}
\end{center}

process
$29+45=x+58$

$16$

}$\angle x$̑傫߂BȂC_O͉~̒SłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic150.tex}
\end{center}

process
$90-15$

$75$

}$\angle x$̑傫߂BȂC_O͉~̒SłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic151.tex}
\end{center}

$154$

}$\angle x$̑傫߂BȂC_O͉~̒SłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic152.tex}
\end{center}

$74$

}̂悤ɁC$\triangle $ABCC̓ډ~ƕABC
BCCCAƂ̐ړ_PCQCRƂB
$\angle$ BAC $ =56$C$\angle$ QPR $ =54$̂ƂC
$\angle $ABC̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic153.tex}
\end{center}

process
ڌ藝C$\angle $ RQC $=54$C\\
$\angle $ QRC $=54$B
$\angle $ ACB $=180-54 \times 2=72$B\\
$\angle $ ABC $=180-(56+72)=52$

$52$

}ɂāC$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic154.tex}
\end{center}

process
}ɂāC$\angle$ABD$=\angle$ACDł邱ƂCACBCCCD4̓_͓~ɂ邱ƂɂȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic154p.tex}
\end{center}

$32$

̐}$\angle x$̑傫߂B
C$\stackrel{\textstyle\frown}{\mathrm{AC}}=\stackrel{\textstyle\frown}{\mathrm{AD}}$
łCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic201.tex}
\end{center}

process
$\stackrel{\textstyle\frown}{\mathrm{AC}}=\stackrel{\textstyle\frown}{\mathrm{AD}}$C
$\angle ADC=60$ƂȂ$\triangle $ACD͐Op`B
܂C$\angle BAD=90$ł邩C
$x=90-60=30$

$30$

̐}$\angle x$̑傫߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic202.tex}
\end{center}

process
AEƁC\\
$x+70=90$$x=20$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic202p.tex}
\end{center}

$20$

̐}$\angle x$̑傫߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic203.tex}
\end{center}

process
$\triangle $OBC͓񓙕ӎOp`ł邩C$\angle OCB=30$B
$\angle DCB=80$ł邩C
$x+30=80$

$50$

̐}$\angle x$̑傫߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic204.tex}
\end{center}

process
$\angle BOCiAƔΑj=220$C$x=360-220=140$

140

}ŁC$\angle x$̑傫߂B
CACB͉~̐ړ_CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic207.tex}
\end{center}

process
}ɂC$\angle DOA=180-90-75=15$C\\
$x=2 \times 15=30$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic207p.tex}
\end{center}
iʁj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic207pp.tex}
\end{center}
ABɑ΂~p$\angle APB$CƔΑɍƁC\\
$\angle APB=180-105=75$\\
$\angle AOB=2 \times \angle APB=150$\\
lp`AOBDɂ\\
$150+90+90+x=360$

$30$

}ŁC$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic209.tex}
\end{center}

process
$\angle$BCD$=x+25$B
܂Cڌ藝ɂ$\angle$BCD$=60$ł邩C
$x+25=60$C$x=35$

$35$

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic401.tex}
\end{center}

process
}C$60+25+x=130$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic401p.tex}
\end{center}

40

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic402.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic402p.tex}
\end{center}

70

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic403.tex}
\end{center}

67

̐}$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic404.tex}
\end{center}

220



[Level2]
p
̐}$\angle x$$\angle y$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic162.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic162p.tex}
\end{center}
BDƁC~p̒藝ɂ$\angle$ABD$=68$B
$x=90-68=22$B
Op̒藝ɂ$x+40=y$B$y=62$

$x=22,\,y=62$

}ŁCAT͓_Aɂ~̐ڐŁC ABC ͉~ɓڂĂB
$\angle$ BAT$=40$CAB:AC$=2:3$łƂC$\angle$ BAC̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic12.tex}
\end{center}

process
$\angle$ C: $\angle$ B$=2:3$B܂Cڌ藝$\angle$ C$=40$B
āC$\angle$ B=$60$B\\
$\angle$ BAC$=180-(40+60)=80$

$80$

} $\angle$ $x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic25.tex}
\end{center}

process
~ɓڂlp`C$\angle$ CDA $=(180-x)$B$\angle$ CDE $=x$B
$\triangle $ FBClƁC\\
$\angle$ FCE $=(x+35)$B\\
$\triangle $ DCEɂāC$x+(x+35)+30=180$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic25p.tex}
\end{center}

$57.5$

}ŁC$\angle x$̑傫߂BCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic205.tex}
\end{center}

process
$\angle FAD=50+35=85$\\
$x+85=\angle ADC$ł邪C
$\angle ADC=130$ł邩C\\
$x+85=130$B$x=45$

$45$

}ŁC$\angle x$̑傫߂B
C_C͉~ƒ$l$̐ړ_łCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic206.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic206p.tex}
\end{center}
BCƁC$\angle ACB=90$Bڌ藝ɂ$\angle ABC=x$B$\angle OAC=90-x$B
}ɂC\\
$40+(90-x)=x$C$x=65$

$65$

}ŁC$\angle x$̑傫߂B
C_B͉~ƒ$l$Ɖ~̐ړ_ŁC
$\stackrel{\textstyle\frown}{\mathrm{AB}}=2\stackrel{\textstyle\frown}{\mathrm{AD}}$
łCO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic208.tex}
\end{center}

process
BDBڌ藝C$\angle$ADB$=x$\\
$\stackrel{\textstyle\frown}{\mathrm{AB}}=2\stackrel{\textstyle\frown}{\mathrm{AD}}$
C
$\angle$ABD$=\displaystyle \frac{1}{2} \angle $ADB$=\displaystyle\frac{1}{2}x$\\
$\triangle $ABD̓p̘aC\\
$75+x+\displaystyle \frac{1}{2}x=180$C$x=70$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic208p.tex}
\end{center}

$70$

}ŁC$\angle x$̑傫߂B
CAB$=$BCC_C͐ړ_łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic210.tex}
\end{center}

process
$\angle$ADC$=180-115=65$B
AB $=$ BCC$\angle$ ADB $=\angle$ BDCB
āC\\
$\angle$BDC$=65\times \displaystyle\frac{1}{2}=32.5$\\
ڌ藝C
$\angle$BCE$=32.5$\\
$x=180-(50+32.5)=97.5$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic210p.tex}
\end{center}

$97.5$

$\triangle$ ABC͓_Cŉ~OɐڂĂBAB͉~O̒aŁCCP͉~O̐ڐłB
$\angle$ BCP$=30$C~O̔a$4cm$̂ƂC$\triangle$ ABC̖ʐς߂B
CO͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic23.tex}
\end{center}

process
ڌ藝C$\angle$ BAC $=30$B$\triangle$ ABC$1:2:\sqrt{3}$̒pOp`łC
CO͉~̒SƂB
BC$=4$CBA$=8$CAC$=4 \sqrt{3}$BāC
$\displaystyle \frac{1}{2} \times 4 \times 4 \sqrt{3}$

$8 \sqrt{3}$






[Level3]
ډ~Ȃ
}ŁC~O͒pOp`ABCɓ_DCECFŐڂĂB̂ƂC~O̔a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic31.tex}
\end{center}

process
ʐςɒڂāC
$ \displaystyle \frac{1}{2} \times 3 \times 4  = \displaystyle \frac{1}{2} x (4+3+5)$

$ x=1 $

}ɂāCOp`ABC̖ʐς$72 cm^2$̂ƂCeӂ̓_DCECFŐڂ~O̔a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic32.tex}
\end{center}

process
CD $=$ CE $ =6 cm $C AF $=$ AE $ =5 cm $ \\
 BF $=7 cm=$ BDB BC $=6+7=13cm$B\\
$ \displaystyle \frac{1}{2}r(11+12+13)=72 $

$4 cm$

}̂悤ȒpOp`ɔ~ڂĂB̔~̔a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic48.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic48p.tex}
\end{center}
pOp`ABC̖ʐς
$\displaystyle \frac{1}{2} \times 6 \times 8=24 cm^2$ \\
$\triangle $ ABO $+\triangle$ACO $=\triangle$ABCB\\
$\displaystyle \frac{1}{2}\times 6 \times r+\displaystyle \frac{1}{2} \times 8 \times r = 24$

$\displaystyle \frac{24}{7}$

̐}͉~̓ڂ}`łB$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic33.tex}
\end{center}

$4 cm$

̐}͉~̓ڂ}`łB$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic34.tex}
\end{center}

5





[Level4]
ʐ
}̎ΐ̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic305.tex}
\end{center}

process
̐}͉}̂悤ɍl邱ƂłB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic305p.tex}
\end{center}
$\displaystyle\frac{1}{2} \times 8\times 4=16\,{\rm cm^2}$

$16\,{\rm cm^2}$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic82.tex}
\end{center}

process
$2 \left( \displaystyle \frac{1}{4} \times \pi \,4^2-\displaystyle \frac{1}{2} \times 4^2 \right)=2(4 \pi-8)$

$8 \pi -16$

}̎ΐ̖ʐρiP$cm^2$j߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic41.tex}
\end{center}

process
$2 \left( \displaystyle \frac{1}{4} \times ~̖ʐ -Op`̖ʐ \right)\\ 
=2 \left( \displaystyle \frac{1}{4} \times \pi E5^2-\displaystyle \frac{1}{2} \times 5^2 \right)$@

$\displaystyle \frac{25}{2} \pi -25 $

}̎ΐ̖ʐρiP$cm^2$j߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic111.tex}
\end{center}

process
$2 \left( \displaystyle \frac{1}{4} \times ~̖ʐ -Op`̖ʐ \right)\\ 
=2 \left( \displaystyle \frac{1}{4} \times \pi E8^2-\displaystyle \frac{1}{2} \times 8^2 \right)$@

$32 \pi -64 $

}̎ΐ̖ʐρiP$cm^2$j߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic54.tex}
\end{center}

process
$2 \left( \displaystyle \frac{1}{4} \times ~̖ʐ -Op`̖ʐ \right)\\ 
=2 \left( \displaystyle \frac{1}{4} \times \pi E6^2-\displaystyle \frac{1}{2} \times 6^2 \right)$@@

$18 \pi -36 $

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic44.tex}
\end{center}

process
iaAC̔~j$-$iaBC̔~j$+$iaAB̔~j
$=\piE5^2 \times \displaystyle \frac{1}{2}-\piE2^2 \times \displaystyle \frac{1}{2}+ \piE3^2 \times \displaystyle \frac{1}{2}$

$15 \pi cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic306.tex}
\end{center}

process
iaAC̔~j$-$iaBC̔~j$+$iaAB̔~j
$=\piE15^2 \times \displaystyle \frac{1}{2}-\piE10^2 \times \displaystyle \frac{1}{2}+ \piE5^2 \times \displaystyle \frac{1}{2}\\
=\displaystyle \frac{\pi}{2}\,(15^2-10^2+5^2)=\displaystyle \frac{\pi}{2}\times 150$

$75 \pi \,cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic45.tex}
\end{center}

process
`ABCD͕H`ƍlƌvZBABCD̖ʐ=AC $ \times $ DB $ \times \displaystyle \frac{1}{2}
=10 \times 10 \times \displaystyle \frac{1}{2}$ B\\
ΐ̖ʐ $=\piE5^2-10^2\times \displaystyle \frac{1}{2}$

$25(\pi-2) =25\pi-50 \,cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic114.tex}
\end{center}

process
`ABCD͕H`ƍlƌvZBABCD̖ʐ=AC $ \times $ DB $ \times \displaystyle \frac{1}{2}
=6 \times 6 \times \displaystyle \frac{1}{2}$ B\\
ΐ̖ʐ $=\piE3^2-6^2\times \displaystyle \frac{1}{2}$

$9(\pi-2) =9\pi-18\, cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic115.tex}
\end{center}

process
`ABCD͕H`ƍlƌvZBABCD̖ʐ=AC $ \times $ DB $ \times \displaystyle \frac{1}{2}
=8 \times 8 \times \displaystyle \frac{1}{2}$ B\\
ΐ̖ʐ $=\piE4^2-8^2\times \displaystyle \frac{1}{2}$

$16\pi-32 cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic53.tex}
\end{center}

process
`ABCD͕H`ƍlƌvZBABCD̖ʐ=AC $ \times $ DB $ \times \displaystyle \frac{1}{2}
=20 \times 20 \times \displaystyle \frac{1}{2}=200$ B\\
ΐ̖ʐ $=\piE10^2-200$

$100(\pi-2)=100\pi-200 cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic46.tex}
\end{center}

process
$\left( \displaystyle\frac{1}{4} \times \pi \cdot 5^2-\displaystyle\frac{1}{2} \times 5^2 \right)\times 8 $

$50\pi-100\, cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic118.tex}
\end{center}

process
$\left( \displaystyle\frac{1}{4} \times \pi \cdot 4^2-\displaystyle\frac{1}{2} \times 4^2 \right)\times 8 $

$32\pi-64\, cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic119.tex}
\end{center}

process
$\left( \displaystyle\frac{1}{4} \times \pi \cdot 6^2-\displaystyle\frac{1}{2} \times 6^2 \right)\times 8 $

$72(\pi-2)=72\pi-144\, cm^2$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic52.tex}
\end{center}

process
$\left\{ \displaystyle\frac{1}{4} \times \pi \cdot \left(  \displaystyle \frac{1}{2}\right)^2-\displaystyle\frac{1}{2} \times \left(  \displaystyle \frac{1}{2}\right)^2 \right\}\times 8 $

$\displaystyle \frac{1}{2}\pi-1$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic49.tex}
\end{center}

process
$\displaystyle \frac{1}{4} \pi E4^2-\displaystyle \frac{1}{2} \pi E2^2 $

$ 2 \pi (cm^2)$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic112.tex}
\end{center}

process
$\displaystyle \frac{1}{4} \pi E8^2-\displaystyle \frac{1}{2} \pi E4^2 $

$ 8 \pi (cm^2)$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic113.tex}
\end{center}

process
$\displaystyle \frac{1}{4} \pi E12^2-\displaystyle \frac{1}{2} \pi E6^2 $

$ 18 \pi (cm^2)$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic50.tex}
\end{center}

process
$\displaystyle \frac{1}{4} \pi E16^2-\displaystyle \frac{1}{2} \pi E8^2 $

$ 32 \pi (cm^2)$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic51.tex}
\end{center}

process
$2 \times \left( \displaystyle \frac{1}{4} \times \pi E14^2-\displaystyle \frac{1}{2} \times 14 \times 14 \right)  \\
=2 (49\pi -98)$

$98 (\pi-2)=98 \pi-196 (cm^2)$

̐}͒a4Ƃ锼~łB
̎ΐ̖ʐς߂ȂB
C~$\pi$C_O͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic85.tex}
\end{center}

process
~$=\displaystyle \frac{1}{2} \times \pi \cdot 2^2=2 \pi$\\
Op`$=\displaystyle \frac{1}{2} \times 4 \times 2 =4$\\
~$-$Op`$=2 \pi-4$

$2 \pi-4$

̐}͒a8Ƃ锼~łB
̎ΐ̖ʐς߂ȂB
C~$\pi$C_O͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic84.tex}
\end{center}

process
~$=\displaystyle \frac{1}{2} \times \pi \cdot 4^2=8 \pi$\\
Op`$=\displaystyle \frac{1}{2} \times 8 \times 4 =16$\\
~$-$Op`$=8 \pi-16$

$8 \pi-16$

̐}͒a12Ƃ锼~łB
̎ΐ̖ʐς߂ȂB
C~$\pi$C_O͉~̒SƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic83.tex}
\end{center}

process
~$=\displaystyle \frac{1}{2} \times \pi \times 6^2=18 \pi$\\
Op`$=\displaystyle \frac{1}{2} \times 12 \times 6 =36$\\
~$-$Op`$=18 \pi-36$

$18 \pi-36$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic106.tex}
\end{center}

process
$\displaystyle \frac{\pi \times 6^2}{2}-\left( \displaystyle \frac{\pi \times 4^2}{2} 
+\displaystyle \frac{\pi \times 2^2}{2}\right)\\
=18\pi -(8\pi +2\pi)=8 \pi$

$8 \pi \, cm^2$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic120.tex}
\end{center}

process
$\displaystyle \frac{\pi \times 7^2}{2}-\left( \displaystyle \frac{\pi \times 4^2}{2} 
+\displaystyle \frac{\pi \times 3^2}{2}\right)\\
=\displaystyle \frac{1}{2} \{ 49\pi -(16\pi +9\pi) \}=12 \pi$

$12 \pi \, cm^2$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic121.tex}
\end{center}

process
$\displaystyle \frac{\pi \times 5^2}{2}-\left( \displaystyle \frac{\pi \times 3^2}{2} 
+\displaystyle \frac{\pi \times 2^2}{2}\right)\\
=\displaystyle \frac{1}{2} \{25\pi -(9\pi +4\pi) \}=6 \pi$

$6 \pi \, cm^2$

}̎ΐ̖ʐς߂BC\\
AB $=4$cmCBC $=$ CD $=2$cmC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic108.tex}
\end{center}

process
$\left( \displaystyle \frac{\pi \times 3^2}{2}-\displaystyle \frac{\pi \times 2^2}{2} \right)
+\left( \displaystyle \frac{\pi \times 2^2}{2}-\displaystyle \frac{\pi \times 1^2}{2} \right)\\
=\displaystyle \frac{1}{2} (9 \pi-4 \pi)+\displaystyle \frac{1}{2} (4 \pi- \pi)\\
=\displaystyle \frac{1}{2} \cdot 5 \pi+\displaystyle \frac{1}{2} \cdot 3 \pi=4\pi$

$4\pi \, cm^2$

}4̉~͓ałB
ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic107.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic107p.tex}
\end{center}
$4^2-\pi \cdot 2^2$

$16-4 \pi \, cm^2$

}4̉~͓ałB
ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic116.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic116p.tex}
\end{center}
$8^2-\pi \cdot 4^2$

$64-16 \pi \, cm^2$

}4̉~͓ałB
ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic117.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic117p.tex}
\end{center}
$6^2-\pi \cdot 3^2$

$36-9 \pi \, cm^2$

}̎ΐ̖ʐς߂BC\\
AB $=20$cmCBC $=40$cmC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic110.tex}
\end{center}

process
$\left( \displaystyle \frac{\pi \times 30^2}{2} -\displaystyle \frac{\pi \times 20^2}{2} \right)
+\displaystyle \frac{\pi \times 10^2}{2}\\
=\displaystyle \frac{\pi}{2} \times(900-400)+\displaystyle \frac{\pi}{2} \times 100
=\displaystyle \frac{\pi}{2}\times (500+100)\\
=300 \pi $

$300 \pi \, cm^2$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic307.tex}
\end{center}

process
$\pi \cdot 3^2+12^2-6^2=\pi \cdot 3^2+(12-6)(12+6)\\ =9\pi+108$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic307p.tex}
\end{center}

$9\pi+108\,cm^2$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic308.tex}
\end{center}

process
$\pi \cdot 1^2+4^2-2^2=\pi +12$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic308p.tex}
\end{center}

$\pi +12\,cm^2$

}̎ΐ̖ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic309.tex}
\end{center}

process
$\pi \cdot 2^2+8^2-4^2=4\pi +48$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic309p.tex}
\end{center}

$4\pi +48\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic311.tex}
\end{center}

process
$4^2-\pi\cdot 2^2$

$16-4 \pi\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic312.tex}
\end{center}

process
$2^2-\pi\cdot 1^2$

$4- \pi\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic313.tex}
\end{center}

process
$8^2-\pi\cdot 4^2$

$64- 16\pi\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic314.tex}
\end{center}

process
$\left(\displaystyle\frac{1}{4}\times \pi\cdot 2^2-\displaystyle\frac{1}{2}\times2^2\right)\times 8\\
=(\pi-2)\times 8$

$8\pi- 16\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic315.tex}
\end{center}

process
$\left(\displaystyle\frac{1}{4}\times \pi\cdot 3^2-\displaystyle\frac{1}{2}\times3^2\right)\times 8\\
=\left(\displaystyle\frac{9}{4}\pi-\displaystyle\frac{9}{2} \right)\times 8$

$18\pi- 36\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic316.tex}
\end{center}

process
$\left(\displaystyle\frac{1}{4}\times \pi\cdot 2^2-\displaystyle\frac{1}{2}\times2^2\right)\times 8\\
=\left(\pi-2 \right)\times 8$

$8\pi- 16\,cm^2$

̐}`̎ΐ̖ʐς߂ȂBC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic317.tex}
\end{center}

process
$\left(\displaystyle\frac{1}{4}\times \pi\cdot 4^2-\displaystyle\frac{1}{2}\times4^2\right)\times 8\\
=\left(4\pi-8 \right)\times 8$

$32\pi- 64\,cm^2$



[Level5]
`̖ʐ
}̎ΐ̖ʐς߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic264.tex}
\end{center}

process
ʐ$S=\pi \times 6^2\times \displaystyle\frac{60}{360}-\pi \times 3^2\times \displaystyle\frac{60}{360}
=\displaystyle\frac{9}{2}\pi$

$\displaystyle\frac{9}{2}\pi \ cm^2$

a̒$8 cm$ŁCSp45̐`̖ʐς߂Bi~$\pi$Ƃj

process
$\pi \times 8^2 \times \displaystyle \frac{45}{360}=8 \pi$

$8 \pi cm^2$

a̒$6 cm$ŁCSp120̐`̖ʐς߂Bi~$\pi$Ƃj

process
$\pi \times 6^2 \times \displaystyle \frac{120}{360}=12 \pi$

$12 \pi cm^2$

}̂悤Ȑ`̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic62.tex}
\end{center}

process
$\pi E5^2 \times \displaystyle \frac{288}{360}=\pi E5^2 \times \displaystyle \frac{4}{5}$

$20 \pi cm^2$

}̐`̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic302.tex}
\end{center}

process
$\pi\times 12^2 \times \displaystyle\frac{70}{360}$

$28\pi\,{\rm cm^2}$

}̐`̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic303.tex}
\end{center}

process
$\pi\times 9^2 \times \displaystyle\frac{240}{360}$

$54\pi\,{\rm cm^2}$









[Level6]
~
}̂悤ȉ~B̉~̑ʂ𕽖ʏɓWJĂł`̒Sp͉xH
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic61.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic61p.tex}
\end{center}
߂钆Sp$a$ƂƁC\\
`̌ǂ̒=ʂ̉~̉~$=2 \times \pi \times 10=20 \pi$B
a$40 cm$̉~̉~́C$2 \times \pi \times 40=80 \pi$B\\
$\displaystyle \frac{20 \pi}{80 \pi}=\displaystyle \frac{a}{360}$B$a =90$

$90$

}̂悤ȉ~̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic63.tex}
\end{center}

process
$\pi \times 4^2+\pi \times 12^2 \times \displaystyle \frac{2 \pi \times 4}{2 \pi \times 12}
=16 \pi + \pi \times 12 \times 4\\
=16 \pi + 48 \pi$\\
iQlja$r$C$l$̉~̑ʐς\\
$S=\pi\times l^2\times\displaystyle\frac{2\pi r}{2\pi l}=\pi rl$ł邩C\\
̖ł́Cʐ$=\pi \times 12 \times 4$

$64\pi cm^2$

}̉~̕\ʐς͂炩BC~ $\pi$ ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic64.tex}
\end{center}

process
$\pi E8^2 \times \displaystyle \frac{2 \pi E4}{2 \pi E8}+\piE4^2$\\
iʉj
a$r$C$l$̉~̑ʐς\\
$S=\pi\times l^2\times\displaystyle\frac{2\pi r}{2\pi l}=\pi rl$ł邩C\\
̖ł́Cʐ$=\pi \times 4 \times 8=32\pi$B\\
߂\ʐ$=32 \pi+\pi \times 4^2$

$48 \pi(cm^2)$

}̉~̕\ʐς͂炩BC~ $\pi$ ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic265.tex}
\end{center}

process
$\pi E5^2 \times \displaystyle \frac{2 \pi E2}{2 \pi E5}+\piE2^2$\\
iʉj
a$r$C$l$̉~̑ʐς\\
$S=\pi\times l^2\times\displaystyle\frac{2\pi r}{2\pi l}=\pi rl$ł邩C\\
̖ł́Cʐ$=\pi \times 2 \times 5=10\pi$B\\
߂\ʐ$=10 \pi+\pi \times 2^2$

$14 \pi(cm^2)$

}͒ʂ̔a$3 cm$C$9 cm$̒~̕ʐ}Ɨʐ}łB~̑ʏPABɎŒZ̐ɂ_ṔC
ʂ̍߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic65.tex}
\end{center}

process
ʐ}CAP$'=2 \pi \times 3 \times \displaystyle \frac{120}{360}=2 \pi$B
āCWJ}̐AP$'=2 \pi$B$\displaystyle \frac{PP'}{AP'}=\displaystyle \frac{BC}{AC}$B
$\displaystyle \frac{PP'}{2 \pi}=\displaystyle \frac{9}{6 \pi}$BPP$'=3$ cm
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic65p.tex}
\end{center}

$3$ cm

~̓WJ}CSp120Ca5cm̐`łƂC
ʂ̔a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic171.tex}
\end{center}

process
߂ʂ̔a̒$r$ƂƁC\\
$2\pi \times 5 \times \displaystyle\frac{120}{360}=2\pi r$\\
$2\pi \times 5 \times \displaystyle\frac{1}{3}=2\pi r$C
$r=\displaystyle\frac{5}{3}$

$\displaystyle\frac{5}{3}$

}́C~̓WJ}łB
ʂ̂`̒Sp$x$͉B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic172.tex}
\end{center}

process
$2\pi \times 9\times \displaystyle\frac{x}{360}=2\pi \times 3$\\
$x=120$

$120$

ʂ̔a$5 {\rm cm}$C̒$13 {\rm cm}$̉~B̉~̕\ʐς߂B
C~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic266.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic266p.tex}
\end{center}
$\pi \times 13^2 \times \displaystyle\frac{2\pi \times 5}{2\pi \times 13}+\pi \times 5^2\\
=\pi \times 13 \times 5+\pi \times 5^2=65\pi+25\pi=90\pi$

$90\pi\, {\rm cm^2}$















[Level7]
~
a̒$4cm$̉~̎̒߂Bi~$\pi$Ƃj

process
$2 \times \pi \times 4=8 \pi$

$8 \pi cm$

a̒$3cm$̉~̎̒߂Bi~$\pi$Ƃj

process
$2 \times \pi \times 3=6 \pi$

$6 \pi cm$

̐}͒a$10 cm$̉~W܂̂ЂŔ̂łB
іڂlȂƂĈЂ̒͂KvH\, C~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic81.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic81p.tex}
\end{center}
$10 \times 4+2 \pi \times 5=40+10 \pi$

$40+10\pi$

̐}͒a$8 cm$̉~W܂̂ЂŔ̂łB
іڂlȂƂĈЂ̒͂KvH\, C~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic81c.tex}
\end{center}

process
$8 \times 4+2 \pi \times 4=32+8 \pi$

$32+8\pi$

̐}͒a$4 cm$̉~W܂̂ЂŔ̂łB
іڂlȂƂĈЂ̒͂KvH\, C~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic81c.tex}
\end{center}

process
$4 \times 4+2 \pi \times 2=16+4 \pi$

$16+4\pi$

}́Cʂ̔a$5 cm$C$10 cm$ł~łB
̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic155.tex}
\end{center}

process
$(\pi \cdot 5^2) \times 2+2 \pi \times 5 \times 10=150 \pi$

$150\pi(cm^2)$

}́Cʂ̔a$4 cm$C$8 cm$ł~łB
̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic155c.tex}
\end{center}

process
$(\pi \cdot 4^2) \times 2+2 \pi \times 4 \times 8=96 \pi$

$96\pi(cm^2)$

}́Cʂ̔a$2 cm$C$4 cm$ł~łB
̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic155d.tex}
\end{center}

process
$(\pi \cdot 2^2) \times 2+2 \pi \times 2 \times 4=24 \pi$

$24\pi(cm^2)$

}́Cʂ̔a$3 cm$C$6 cm$ł~łB
̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic155e.tex}
\end{center}

process
$(\pi \cdot 3^2) \times 2+(2 \pi \times 3) \times 6=54 \pi$

$54\pi(cm^2)$

}̎ΐ̎̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic264.tex}
\end{center}

process
̒$l=2\times \pi\times 3\times\displaystyle\frac{60}{360}+\\
2\times \pi\times 6\times\displaystyle\frac{60}{360}+3\times 2=3\pi +6$

$(3\pi +6) \ cm$

}̐`̌ʂ̒߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic302.tex}
\end{center}

process
$2\pi\times 12 \times \displaystyle\frac{70}{360}$

$\displaystyle\frac{14}{3}\,\pi$cm

}̐`̌ʂ̒߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic303.tex}
\end{center}

process
$2\pi\times 9 \times \displaystyle\frac{240}{360}$

$12\pi\,{\rm cm}$

ʂ̔a3cmC4cm̉~̕\ʐς߂ȂBC~$\pi$ƂB

process
$2(\pi \times 3^2)+(2\pi \times 3)\times 4=42 \pi$

$42\pi\,cm^2$

ʂ̔a2cmC3cm̉~̕\ʐς߂ȂBC~$\pi$ƂB

process
$2\,(\pi \times 2^2)+(2\pi \times 2)\times 3=20 \pi$

$20\pi\,cm^2$

ʂ̔a4cmC5cm̉~̕\ʐς߂ȂBC~$\pi$ƂB

process
$2\,(\pi \times 4^2)+(2\pi \times 4)\times 5=72 \pi$

$72\pi\,cm^2$







[Level8]
ׂ̒藝C~Ƒ
}$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic251.tex}
\end{center}

process
$x \times 8=5 \times 6$C$x=\displaystyle \frac{15}{4}$

$\displaystyle \frac{15}{4}$

}$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic252.tex}
\end{center}

process
$2 \times 6=3 \times (3+x)$C$x=1$

$1$

}$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic253.tex}
\end{center}

process
$3 \times (3+x)=5^2$C$x=\displaystyle \frac{16}{3}$

$\displaystyle \frac{16}{3}$

}ŁC$\triangle$ ABC~ɓڂĂB_Bɂ邱̉~̐ڐAC̉Ƃ̌_PƂB
BC $=$ BP $=7cm$CAP $=4cm$̂ƂCAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic24.tex}
\end{center}

process
AC$=x$ƂƁCׂ̒藝C\\
$4 \times (x+4)=7^2$C$x=\displaystyle\frac{33}{4}=8.25$\\
iʁjڌ藝C$\angle$ ABP $= \angle$ ACBB܂CBC=BPC$\angle$ BCA = $\angle$ BPAB\\
 $\angle$ ABP = $\angle$ APBB\\
$\triangle $APB  $\triangle $BCPBAC$=x$ƂƁC\\
$\displaystyle \frac{4+x}{7}=\frac{7}{4}$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic24p.tex}
\end{center}

$8.25$

̐}ɂāC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic261.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic261p.tex}
\end{center}
$\triangle$ABE$ \triangle$DCEC\\
$6:4=3:x$C$x=2$

$x=2$

̐}ɂāC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic262.tex}
\end{center}

process
ׂ̒藝C\\
$x\times (x+5)=4\times 6$C$x^2+5x-24=0$C\\
$(x+8)(x-3)=0$C$x=3$i$x>0$j\\
iʁj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic262p.tex}
\end{center}
$\triangle$ABC $\triangle$ADEC\\
$(4+2):x=(x+5):4$

$3$

̐}ɂāC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic263.tex}
\end{center}

process
ׂ̒藝C\\
$3\times (3+x)=4^2$C$x=\displaystyle\frac{7}{3}$\\
iʁj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic263p.tex}
\end{center}
$\triangle$ABC $\triangle$BDCC\\
$(x+3):4=4:3$C$3(x+3)=16$

$x=\displaystyle\frac{7}{3}$




[EOF]
% t@C̍Ōɂ
