% %ȉ̕RgƂ̂łC_ł͂܂ł܂B
% ̃^Cg
[Title]
401_Op`̐

% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
% tHg̑傫B1`10C܂TeX̃R}hw肷B
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% vAuɒǉpbP[Wt@Cw肷B
[usepackage]
\usepackage{color}

% ꂼ̖𓚂$\displaystyle $tꍇ́C@ON ܂1
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% @@@@@@@@@@@@@@@@@@@ɂݒ肪D悳܂B
[displaystyle]
OFF


% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
px߂
̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic53.tex}
\end{center}

135

̐}$\angle x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic54.tex}
\end{center}

125

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic55.tex}
\end{center}

process
lp`̓p̘a$180 \times 2=360$Ȃ̂ŁC\\
$\angle x=360-(80+95+100)=360-275=85$

85

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic56.tex}
\end{center}

130

̐}$\angle x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic57.tex}
\end{center}

110

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic58.tex}
\end{center}

process
$180-x=360-(65+65+140)$B$x=90$

90

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic18.tex}
\end{center}

85

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic19.tex}
\end{center}

145

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic20.tex}
\end{center}

50

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic71.tex}
\end{center}

process
u[^B$40+30+35=105$

105

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic72.tex}
\end{center}

80

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic01.tex}
\end{center}

120

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic02.tex}
\end{center}

100

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic03.tex}
\end{center}

40

}́$x$̑傫߂Bi$l$C$m$C$n$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic04.tex}
\end{center}

72

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic05.tex}
\end{center}

120

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic06.tex}
\end{center}

55

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic07.tex}
\end{center}

process
u[^B$45+35+30$

110

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic08.tex}
\end{center}

30

}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic09.tex}
\end{center}

70

̐}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic10.tex}
\end{center}

110

̐}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic11.tex}
\end{center}

process
$100+110+152+67+x=180 \times 3$

111

̐}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic12.tex}
\end{center}

80

̐}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic13.tex}
\end{center}

65

}$l$//$m$̂ƂC$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic14.tex}
\end{center}

70

̐}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic15.tex}
\end{center}

process
$80+70+60+90+x=360$

60

̐}́$x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic16.tex}
\end{center}

process
$60+55+55+65+50+x=360$

75

̐}́$x$̑傫߂BClp`ABCD͕slӌ`łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic17.tex}
\end{center}

25

}$xCy$̒l߂ȂBC$m//n$ƂB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic52.tex}
\end{center}

$x=67Cy=106$

}̂悤ȁC5̒_}`̓p̘a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic73.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic73p.tex}
\end{center}
a$+$b$+$c$+$d$+$e$=180$

180

}̂悤ɁC3̒1_ŌĂB
̂ƂC$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic81.tex}
\end{center}

$95$

̐}$\angle x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic82.tex}
\end{center}

$90$

̐}$\angle x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic83.tex}
\end{center}

$130$

}̎lp`ABCD͕slӌ`łB
$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic84.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic84p.tex}
\end{center}
$20+x=60$

$40$

̐}$\angle x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic85.tex}
\end{center}

process
$120-36=84$

$84$

}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic86.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic86p.tex}
\end{center}

$60$

}́$x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic87.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic87p.tex}
\end{center}

$65$

}̎lp`ABCD͕slӌ`łBAD$=$DÊƂC$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic88.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic88p.tex}
\end{center}

$32$

}̎lp`ABCD͕slӌ`łB$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic89.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic89p.tex}
\end{center}

$30$

̐}$\angle x$̑傫߂Bi$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic96.tex}
\end{center}

24

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic97.tex}
\end{center}

130

̐}$\angle x$̑傫߂BiC$l$C$m$͕sj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic98.tex}
\end{center}

process
}C$x+50=85^{\circ}$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic98p.tex}
\end{center}

35

̐}$\angle x$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic99.tex}
\end{center}

140

̐}$l // m$̂ƂC$\angle x$̑傫߂ȂB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic302.tex}
\end{center}

process
$100+45=145$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic302p.tex}
\end{center}

145






[Level2]
ӂ̔
̐}$x$̑傫߂BiDE//BCj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic59.tex}
\end{center}

process
$\displaystyle \frac{x}{4}=\displaystyle \frac{9}{6}$

6

̐}$x$̑傫߂BiAD//EF//BCj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic60.tex}
\end{center}

process
$\displaystyle \frac{x}{5.4}=\displaystyle \frac{8}{6}$

7.2

̐}$x$̑傫߂BiDE//BCj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic61.tex}
\end{center}

process
$\displaystyle \frac{x}{12}=\displaystyle \frac{6}{8}$

9

̐}$x$̑傫߂BiAD//EF//BCj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic62.tex}
\end{center}

process
$\displaystyle \frac{x}{7.2}=\displaystyle \frac{5}{6}$

6

}$DE//BC$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic21.tex}
\end{center}

process
$\displaystyle \frac{x}{3}=\frac{4}{2}$

$6$

}$BC//DE$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic22.tex}
\end{center}

process
$\displaystyle \frac{x}{9}=\frac{4}{12}$

3

}$l$//$m$//$n$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic23.tex}
\end{center}

process
$\displaystyle \frac{x}{12}=\frac{6}{9}$

$8$

}$DE$//$BC$̂ƂC$x$C$y$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic24.tex}
\end{center}

process
$\displaystyle \frac{y}{8}=\frac{6}{10}$\\
$\displaystyle \frac{x}{4.5}=\frac{10}{6}$

$x=7.5Cy=4.8$

}$l$//$m$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic25.tex}
\end{center}

process
$\displaystyle \frac{x}{6}=\frac{6}{4}$\\
$\displaystyle x=6 \times \frac{6}{4}$

$x=9$

}$l$//$m$//$n$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic26.tex}
\end{center}

process
$\displaystyle \frac{x}{18-x}=\frac{14}{7}$

$x=12$

}$l$//$m$//$n$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic27.tex}
\end{center}

process
$\displaystyle \frac{x}{12}=\frac{6}{9}$

$x=8$

}$l$//$m$//$n$̂ƂC$y$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic28.tex}
\end{center}

process
$\displaystyle \frac{y}{2}=\frac{6}{3}$

$y=4$

}$l$//$m$//$n$̂ƂC$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic29.tex}
\end{center}

process
}̂悤ɕ⏕āC\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic30.tex}
\end{center}
$\displaystyle \frac{18-x}{12-x}=\frac{32}{12}$\\
$3(18-x)=8(12-x)$\\
$x=8.4$

$x=8.4$

̐}ɂāC$PA$//$QB$//$RC$C$PA=12 cm$C$QB=4 cm$łƂC$RC$̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic46.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic46p.tex}
\end{center}
$\displaystyle \frac{QB}{PA}=\frac{1}{3}C\displaystyle \frac{CB}{CA}=\frac{1}{3}$ \\
$\displaystyle \frac{AB}{AC}=\frac{2}{3}, \displaystyle \frac{BQ}{CR}=\frac{2}{3} $
$\displaystyle \frac{CR}{BQ}=\frac{3}{2},\displaystyle \frac{CR}{4}=\frac{3}{2} $ \\
$\displaystyle CR=4 \times \frac{3}{2}=6 cm$ 

$6 cm$

̐}$x$̑傫߂BiDE//BCj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic63.tex}
\end{center}

process
$\displaystyle \frac{x}{6}=\displaystyle \frac{10}{4}$

15

̐}$x$̑傫߂Bi$\angle$ ADE $=\angle$ ACBj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic64.tex}
\end{center}

process
$\displaystyle \frac{x}{6}=\displaystyle \frac{4}{3}$

8

̐}$x$̑傫߂BiAD//EF//BCCAE$=$EBj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic65.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic65p.tex}
\end{center}
$\displaystyle \frac{x-3}{4}=\displaystyle \frac{1}{2}$

5




[Level3]
O̒藝
̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic801.tex}
\end{center}

$\sqrt{5}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic802.tex}
\end{center}

process
$\sqrt{9^2+6^2}=\sqrt{(3\cdot 3)^2+(3\cdot 2)^2}=3\sqrt{13}$

$3\sqrt{13}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic803.tex}
\end{center}

process
$\sqrt{11^2-7^2}=\sqrt{(11-7)(11+7)}\\ =\sqrt{2\times 2\times 2\times 9}=6\sqrt{2}$

$6\sqrt{2}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic804.tex}
\end{center}

process
$\sqrt{(\sqrt{6})^2-(\sqrt{3})^2}=\sqrt{3}$

$\sqrt{3}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic805.tex}
\end{center}

process
$\sqrt{6^2-(2\sqrt{3})^2}=\sqrt{48}=4\sqrt{3}$

$4\sqrt{3}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic807.tex}
\end{center}

process
$\sqrt{8^2-6^2}=\sqrt{(8-6)(8+6)}\\ =\sqrt{2\times 2\times 7}=\sqrt{28}=2\sqrt{7}$

$2\sqrt{7}$

Ђ`ABCD̑Ίp̒${\rm AC=4cm}$
C${\rm BD=6cm}$łBAB̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic703.tex}
\end{center}

process
$\sqrt{3^2+2^2}=\sqrt{13}$

$\sqrt{13}$cm

}ŕBC̓_ʂAG֍sŒZ\fbox{iIj}cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic103.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic103p.tex}
\end{center}
}C
$AG^2=5^2+4^2=41$C$AG=\sqrt{41}$

$\sqrt{41}$

}$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic41.tex}
\end{center}

process
ABC$1:1:\sqrt{2}$̒pOp`C$AC=6\sqrt{2}$B\\
ACD$1:2:\sqrt{3}$̒pOp`C\\
$\displaystyle \frac{x}{6\sqrt{2}}=\frac{2}{\sqrt{3}}$

$x=4\sqrt{6}$

}$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic42.tex}
\end{center}

process
ABCɂāC$AC^{2}=5^{2}+4^{2}=41$c\textcircled{\small 1} \\
ACDɂāC$AC^{2}=6^{2}+x^{2}$@c\textcircled{\small 2} \\
\textcircled{\small 1}C\textcircled{\small 2}$41=6^{2}+x^{2}@x=\sqrt{5}$

$x=\sqrt{5}$

}$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic43.tex}
\end{center}

process
$AB:BC=30:18=5:3$CABC͕ӂ̔䂪$5:3:4$̒pOp`ł邩C$CA=24$B
ƁC$CD:CA=26:24=13:12$ƂȂ̂ŁCCDA͂Rӂ̒C$5:12:13$̒pOp`BāC\\
$\displaystyle \frac{x}{24}=\frac{5}{12}$

$x=10$

}$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic45.tex}
\end{center}

process
$BD=\sqrt{3^{2}+5^{2}}=\sqrt{34}$ \\
$x=\sqrt{(\sqrt{2})^{2}+(\sqrt{34})^{2}}=\sqrt{2+34}=\sqrt{36}=6$ 

$6 cm$

}̂悤Ȓ̂ɂāC_AD܂Ŏ𒣂ƂCŒẐƂ̎̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic47.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic47p.tex}
\end{center}
${\rm AD}=\sqrt{{16}^2+(16\times 2)^2}=16\sqrt{1+2^2}=16\sqrt{5}$

$16 \sqrt{5}$

}̂悤Ȓ̂Ca $=4$Cb$=3$Cc $=3$̂ƂCAG̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic48.tex}
\end{center}

process
EG $^2=\sqrt{3^2+4^2}=\sqrt{25}=5$B\\
AG $^2=\sqrt{3^2+5^2}=\sqrt{34}$B

$\sqrt{34}$

}̂悤Ȓ̂̑ΊpXY̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic205.tex}
\end{center}

process
$XY^2=XZ^2+ZY^2=3^2+ZY^2$\\
ŁC$ZY^2=4^2+2^2$ł邩C\\
$XY^2=3^2+4^2+2^2=29$C$XY=\sqrt{29}$

$\sqrt{29}$

}̂悤ȉ~B$x$̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic49.tex}
\end{center}

process
\begin{eqnarray*}
 x &=& \sqrt{4^2+(2 \sqrt{2})^2}=\sqrt{16+4 \times 2}=\sqrt{24}\\
 &=& \sqrt{2^2 \times 6}
\end{eqnarray*}

$2 \sqrt{6} cm$

}́C̒$20 cm$ŁCΊpBD$8 cm$̂Ђ`ABCDłB
̂ƂCΊpAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic74.tex}
\end{center}

process
AB$=5$CBO$=4$C$\triangle $ABO$3:4:5$̒pOp`CAO$=3$B

6

}ɂāCAB$=10m$C$\angle $DAB$=30$C$\angle$DBC$=45$̂ƂC
DC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic91.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic91p.tex}
\end{center}
$\displaystyle \frac{x}{1}=\displaystyle \frac{10}{\sqrt{3}-1}$\\
$x=5\left( \sqrt{3}+1 \right)$

$5\left( \sqrt{3}+1 \right)$\ (m)

$\angle B=90$ł钼pOp`ABCɂāC
AB$=12$cmCBC$=16$cmłB
BACɈ̑HƂƂCCH̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic102.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic102p.tex}
\end{center}
$\displaystyle \frac{x}{16}=\displaystyle\frac{16}{20}$C
$x=16\times \displaystyle\frac{16}{20}=12.8$

$12.8$cm

}̂悤ɁCa$4\,cm$̉~OƁCa$6\,cm$̉~O$'$݂ɊOڂĂƂC
ʐڐAB̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic93.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic93p.tex}
\end{center}
}ɂāC\\
AB$=$OH$=\sqrt{10^2-2^2}=\sqrt{(10-2)(10+2)}\\ =\sqrt{4\times 2\times 4\times 3}=\sqrt{16\times 6}=4\sqrt{6}$

$4\sqrt{6}$\, (cm)

}̂悤ȒABCD$-$EFGHB_ABF̓_ʂē_GɂŒZ߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic101.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic101p.tex}
\end{center}
߂ŒZ$x$ƂƁC\\
$x=\sqrt{6^2+8^2}=\sqrt{100}=10$

$10$

}̒`ɂāC
AB$=15$CBD$=30$ƂB
BD$1:2$̔ɕ_EɂBDɐȐ_FƂB
EF̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic206.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic206p.tex}
\end{center}
$AB:BD=1:2$C$\angle BDA =30$C\\
$AD=15\sqrt{3}$B
$\displaystyle\frac{EF}{BE}=\displaystyle\frac{1}{\sqrt{3}}$ł邩C\\
$EF=10\times \displaystyle\frac{1}{\sqrt{3}}=\displaystyle\frac{10}{3}\sqrt{3}$

$\displaystyle\frac{10}{3}\sqrt{3}$

Oڂ2~OCO$'$̔aꂼ6cmC5cm̂ƂCʐڐPQ̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic207.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic207p.tex}
\end{center}
O$'$OPɉ낵̑HƂƁC\\
OH$=6-5=1$cmCOO$'=6+5=11$cmC\\
O$'$H$=\sqrt{11^2-1^2}=2\sqrt{30}$

$2\sqrt{30}$

}̂悤$\angle$C$=\angle$R̒pOp`ɂāCAB$=5cm$C
AC$:$BC$=1:2$B̂ƂCCABɉ낵̑HƂB
CH̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic208.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic208p.tex}
\end{center}
$\displaystyle\frac{AC}{AB}=\displaystyle\frac{1}{\sqrt{5}}$
C$AC=AB\times\displaystyle\frac{1}{\sqrt{5}}=5\times\displaystyle\frac{1}{\sqrt{5}}=\sqrt{5}$
$\displaystyle\frac{HC}{AC}=\displaystyle\frac{2}{\sqrt{5}}$\\
C$AC=AC\times\displaystyle\frac{2}{\sqrt{5}}=\sqrt{5}\times\displaystyle\frac{2}{\sqrt{5}}=2$

$2$

}ɂāC${\rm BC=6\,cm}$̂ƂC${\rm CD}$̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic9301.tex}
\end{center}

process
${\rm AC}=\displaystyle\frac{6}{\sqrt{3}}$\\
${\rm CD}=\displaystyle\frac{6}{\sqrt{3}}\times \sqrt{2}=2\sqrt{6}$

$2\sqrt{6}$

}́C${\rm \angle BAC}={90}^\circ$C${\rm AB=4\,cm}$C${\rm AC=3\,cm}$C
${\rm BE=6\,cm}$̎OpABC-DEFŁC_P͕BE̓_ƂB̂ƂC${\rm AP+PF}$̒łȂƂC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic9701.tex}
\end{center}

process
$\triangle {\rm ABC}$$3:4:5$̒pOp`ł邩C${\rm BC}=5{\rm cm}$\\
${\rm AF}=\sqrt{9^2+6^2}=3\sqrt{13}$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic9701p.tex}
\end{center}

$3\sqrt{13}$

}́C${\rm \angle BAC}={90}^\circ$C${\rm AB=4\,cm}$C${\rm AC=3\,cm}$C
${\rm BE=3cm}$̎OpABC-DEFŁC_P͕BE̓_ƂB̂ƂC${\rm AP+PF}$̒łȂƂC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic9702.tex}
\end{center}

process
$\triangle {\rm ABC}$$3:4:5$̒pOp`ł邩C${\rm BC}=5{\rm cm}$\\
${\rm AF}=\sqrt{9^2+3^2}=3\sqrt{10}$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic9702p.tex}
\end{center}

$3\sqrt{10}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic821.tex}
\end{center}

process
$\sqrt{7^2-5^2}=\sqrt{(7+5)\cdot (7-5)}\\ =\sqrt{12\cdot 2}=2\sqrt{6}$

$2\sqrt{6}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic822.tex}
\end{center}

process
${\rm OH=9}$C${\rm AH=12}$ł邩C$3:4:5$̎Op`łB

$15$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic824.tex}
\end{center}

process
$\sqrt{7^2-3^2}=\sqrt{4\cdot 10}=2\sqrt{10}$

$2\sqrt{10}$

̒$x$߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic825.tex}
\end{center}

process
$\sqrt{7^2-1^2}=\sqrt{6\cdot 8}=4\sqrt{3}$

$4\sqrt{3}$






[Level4]
Op`̖ʐ
}$AD$$BC$͒płBABC̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic44.tex}
\end{center}

process
\begin{eqnarray*}
 AD &=& \sqrt{17^{2}-15^{2}}=\sqrt{(17-15)(17+15)}\\
 &=& \sqrt{2\times32}=\sqrt{64}=8 \\
\end{eqnarray*}
ABC̖ʐς́C\\
$\displaystyle \frac{1}{2} \times 21 \times 8 = 84 cm^{2}$

$84 cm^{2}$

}ɂāC$\triangle$AOC̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic77.tex}
\end{center}

process
OA$=5$B$\triangle$OAC$1:2:\sqrt{3}$̒pOp`C\\
AC$=5 \sqrt{3}$B$\displaystyle \frac{1}{2} \times 5 \times 5 \sqrt{3}=\displaystyle \frac{25}{2} \sqrt{3}$

$\displaystyle \frac{25}{2} \sqrt{3}$

}ɂāCO͔~̒SŁCCAAړ_ƂڐłB
$\angle$OCA$=30$ŁC~̔a$6$cmłƂC$\triangle $OAB̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic90.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic90p.tex}
\end{center}
$\angle$OAC$=90$C$\angle$OCA$=30$C$\angle$COA$=60$B
OA$=$OBC$\angle$OAB$=\angle $OBABāC$\triangle$OAB͐Op`ƂȂB
$\triangle$OAB̖ʐς\\
$\displaystyle \frac{1}{2} \times 6 \times 3\sqrt{3}=9\sqrt{3}$

$9\sqrt{3}(cm^2)$

}̂悤ȐZp`B
1ӂ̒12cm̂ƂC}̎ΐ̖ʐς߂B
O͐Zp`̒SłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic106.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic106p.tex}
\end{center}
$HA=6,\,OH=6\sqrt{3}$\\
$\triangle$OHA̖ʐ$=\displaystyle\frac{1}{2} \times 6 \times 6\sqrt{3}=18\sqrt{3}$\\
ΐ̖ʐ$=3\times 18\sqrt{3}=54\sqrt{3}$

$54\sqrt{3}$

}̂悤6̐`łł`̒$\triangle$ABCꍇC
`$\triangle$ABC̖ʐς̔߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic202.tex}
\end{center}

process
`\6̐`1ӂ̒1ƂƁC`̖ʐς
$2 \times 3=6$B\\
$\triangle$ABC̖ʐς\\
$6-\left( \displaystyle \frac{1}{2} \times 1 \times 2 \right)
-\left( \displaystyle \frac{1}{2} \times 1 \times 2 \right)
-\left( \displaystyle \frac{1}{2} \times 1 \times 3 \right)
=\displaystyle \frac{5}{2}$\\
āC`$\triangle$ABC̖ʐς̔
$6:\displaystyle \frac{5}{2}=12:5$

`̖ʐρF$\triangle$ABC̖ʐ$=12:5$




[Level5]
`C`̖ʐ
ꂪ4Cꂪ6C5̑`̖ʐς߂B

process
$\displaystyle \frac{1}{2} \times 5 \times (4+6)=25$

25

ꂪ2Cꂪ8C6̑`̖ʐς߂B

process
$\displaystyle \frac{1}{2} \times 6 \times (2+8)=30$

30

}̂悤Ȑ`̓ynB
cC̕ӂɂꂼꕽsɕ$2m$C$4m$̓ĎcԒdɂCԒd̖ʐς$440m^2$ɂȂB
`1ӂ̒߂ȂB\\
A $16m$ B $18m$ C $20m$ D $22m$ E $24m$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic51.tex}
\end{center}

process
$\,(x-2)(x-4)=440$\, \,$x^2-6x-432=0$\\ \,$(x-24)(x+18)=0$ \, $x=24,-18$

E $ 24m$

}̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic50.tex}
\end{center}

process
`ƎOp`ɕčlB\\
$\displaystyle \frac{1}{2} \times 24 \times (20+27)+\displaystyle \frac{1}{2} \times 15 \times 20$

$714cm^2$

}̂悤ɁC`1ӏɂ̕ӂ̗[瓙ɂ2_Ƃđ`B
`̖ʐςC̒`̖ʐς$\displaystyle\frac{3}{5}$ɂȂƂC
}$a$$b$̒̔߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic203.tex}
\end{center}

process
`[2̎Op`̘a$=a\times c$\\
`̖ʐ$=(2a+b) \times c$\\
ӂC\\
$\displaystyle \frac{a\times c}{(2a+b) \times c}=\displaystyle \frac{a}{2a+b}
=\displaystyle \frac{2}{5}$\\
$2(2a+b)=5a$C$2b=a$C$a:b=2:1$

$a:b=2:1$

}ɂāC`ABCD̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic75.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic75p.tex}
\end{center}
EC$=3$B$\triangle$DEC$3:4:5$̒pOp`CDE$=4$B\\
$\displaystyle \frac{1}{2} \times 4 \times (2+5)=14$

14

}̎ΐ̖ʐς߂B
C~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic301.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic301p.tex}
\end{center}
$\displaystyle\frac{1}{2}\times 2\times 8+\displaystyle\frac{1}{2}\times 3\times 4=14$

14



[Level6]
p
}ɂāC\\
$\rm{PA} // \rm{QB} // \rm{RC}$C$\rm{PA}=12 cm,\,\rm{QB}=4\,cm$ łƂC
RC̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic104.tex}
\end{center}

process
$\displaystyle\frac{\rm{QB}}{\rm{CA}}=\displaystyle\frac{1}{3}$C
$\displaystyle\frac{\rm{CB}}{\rm{CA}}=\displaystyle\frac{1}{3}$\\
$\displaystyle\frac{3}{2}=\displaystyle\frac{\rm{AC}}{\rm{AB}}=\displaystyle\frac{\rm{CR}}{\rm{BQ}}=\displaystyle\frac{\rm{CR}}{4}$\\
$\rm{CR}=4\times\displaystyle\frac{3}{2}=6\,$cm

$6$\,cm

}ŁC$\triangle$ABC̍ADƖʐς߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic201.tex}
\end{center}

process
BD$=x$ƂƁC\\
AD$^2=16-x^2$B
܂CDC$=6-x$ł邩C\\
$(6-x)^2+16-x^2=5^2$C$x=\displaystyle\frac{9}{4}$\\
AD$^2=4^2-\left(\displaystyle\frac{9}{4}\right)^2
=\left( 4-\displaystyle\frac{9}{4} \right)\left( 4+\displaystyle\frac{9}{4} \right)$\\
AD$=\displaystyle\frac{5}{4} \sqrt{7}$\\
ʐς\\
$\displaystyle\frac{1}{2} \times 6 \times \displaystyle\frac{5}{4} \sqrt{7}
=\displaystyle\frac{15}{4} \sqrt{7}$

AD$=\displaystyle\frac{5}{4} \sqrt{7}$Cʐς$\displaystyle\frac{15}{4} \sqrt{7}$

}̂悤Ȑ`̐܂莆B
ɔ̐܂ڂC
ɐ}̂悤ɐ܂ƂC$a$̊px߂B
inE14Nxj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic204.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic204p.tex}
\end{center}
PB$=$PC$=$BC$\triangle$PBC͐Op`ł邩C\\
$\angle$PCB$=60$C$\angle$PCD$=30$\\
$\angle$PCE$=\displaystyle\frac{1}{2} \angle$PCD
$=\displaystyle\frac{1}{2} \times 30=15$\\
$\triangle$PCEɂ\\
$a+90+15=180$C$a=75$

$75$

}́C$4 cm$̎e[vABŐ܂Ȃ̂łB
$\angle$ACB$=45$̂ƂC$\triangle$ACB̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic92.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic92p.tex}
\end{center}
}ɂāCCA$=$DACCB$=$DBłCAD//CBCAC//DBł邩C
lp`ACBD͂Ђ`łBāCCA$=$CBB
$\triangle$ACHɂāCAH:CA$=1:\sqrt{2}$ł邩C
AC$=4\sqrt{2}$B\\
CB$=4\sqrt{2}$B߂ʐς́C\\
$\displaystyle \frac{1}{2}\times  4\sqrt{2} \times 4=8 \sqrt{2}$

$8 \sqrt{2}\,(cm^2)$

}1ӂ̒$8cm$̐`łC_ABC̓_Mɏd˂悤
܂Ȃ̂łBBM$=4 cm$̂ƂCPB̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic76.tex}
\end{center}

process
PB$=x$ƂƁCAP$=8-x$BMP$=8-x$BO̒藝ɂC\\
$x^2+4^2=(8-x)^2$B$x=3$

3

}̎Op`ɂāCAC̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic94.tex}
\end{center}

process
}̂悤ȓ_DƂƁC\\
$\rm{BC}:\rm{DC}=2:\sqrt{3}$C$\rm{DC}=5\sqrt{3}$\\
$\rm{CD}:\rm{AC}=1:\sqrt{2}$C$\rm{AC}=5\sqrt{6}$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic94p.tex}
\end{center}

$5\sqrt{6}$

$\rm{AB}=3\rm{cm}$C$\rm{BC}=6\rm{cm}$̒`ABCDEF܂ڂƂāC
CAdȂ悤ɐ܂ƂCBE̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic95.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/401_Op`̐/pic95p.tex}
\end{center}
BE$=x$ƂC$\triangle$ABEɂĎO̒藝pƁC\\
$x^2+3^2=(6-x)^2$\\
$\displaystyle\frac{9}{4}$

$\displaystyle\frac{9}{4}$


[Level7]
O̒藝܂މ~̖ʐ
}ɂāC1/4~ɂQ̔~ڂĂBΐ̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic42.tex}
\end{center}

process
P̔~̔a$x$ƂB
$ \triangle $OPQŎO̒藝pƁC
$(6-x)^2+3^2=(x+3)^2 $B$x=2$B\\
$ \displaystyle \frac{1}{4} \pi \times 6^2- \displaystyle \frac{1}{2}(\pi \times 3^2+ \pi \times 2^2) 
= \displaystyle \frac{18}{2} \pi-\displaystyle \frac{13}{2} \pi= \displaystyle \frac{5}{2} \pi $

$\displaystyle \frac{5}{2} \pi$

ΐ̖ʐς߂Bi~ɂ$\pi$pj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic43.tex}
\end{center}

process
AB͉~̒aȂ̂ŁC$\angle$ BCA $= 90$B\\
CA$=\sqrt{6^2-2^2}=4 \sqrt{2} $B\\
$\triangle $AOC $= \triangle $ ABC $ \times \displaystyle \frac{1}{2}
=\displaystyle \frac{1}{2} \times 2 \times 4 \sqrt{2} \times  \displaystyle \frac{1}{2}$

$2 \sqrt{2}$

}̎ΐ̖ʐς߂BCABC͔a$6 cm$̐`łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic47.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic47p.tex}
\end{center}
`̖ʐ$=\pi \times 6^2 \times \displaystyle \frac{120}{360}=12 \pi$B\\
$\triangle$ ABC̖ʐ $=\displaystyle \frac{1}{2} \times 6 \sqrt{3} \times 3=9 \sqrt{3}$B

$12\pi-9\sqrt{3}$

}̎ΐ̖ʐς߂ȂBiC~$\pi$ƂBj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic55.tex}
\end{center}

process
$(60̐`) \times 2-(6C3\sqrt{3}̎Op`)\\
=\left( \pi \times 6^2 \times \displaystyle \frac{60}{360} \right)\times 2
-\displaystyle \frac{1}{2} \times 6 \times 3 \sqrt{3}$

$12 \pi-9\sqrt{3} cm^2$

}̎ΐ̖ʐς߂BC\\
AB $=2$cmCAC $=4$cmC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/402_~Ɛ`/pic109.tex}
\end{center}

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

$4 \, cm^2$



[EOF]
% t@C̍Ōɂ
