Задание 10, стр 42

Question

AB – общая касательная окружностей с центром O и P, которые касаются
извне. Найдите длину отрезка AB.

ikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt

\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,441); %set diagram left start at 0, and has height of 441

%Flowchart: Connector [id:dp04737421128015451] 
\draw  [color={rgb, 255:red, 189; green, 16; blue, 224 }  ,draw opacity=1 ] (49,126.5) .. controls (49,90.88) and (77.43,62) .. (112.5,62) .. controls (147.57,62) and (176,90.88) .. (176,126.5) .. controls (176,162.12) and (147.57,191) .. (112.5,191) .. controls (77.43,191) and (49,162.12) .. (49,126.5) -- cycle ;
%Flowchart: Connector [id:dp31555659070414244] 
\draw  [color={rgb, 255:red, 189; green, 16; blue, 224 }  ,draw opacity=1 ] (166,85.88) .. controls (166,73.17) and (176.07,62.88) .. (188.5,62.88) .. controls (200.93,62.88) and (211,73.17) .. (211,85.88) .. controls (211,98.58) and (200.93,108.88) .. (188.5,108.88) .. controls (176.07,108.88) and (166,98.58) .. (166,85.88) -- cycle ;
%Flowchart: Connector [id:dp4791347922901814] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (108.88,126.5) .. controls (108.88,124.77) and (110.05,123.37) .. (111.5,123.38) .. controls (112.95,123.38) and (114.13,124.77) .. (114.12,126.5) .. controls (114.12,128.23) and (112.95,129.63) .. (111.5,129.62) .. controls (110.05,129.62) and (108.87,128.23) .. (108.88,126.5) -- cycle ;
%Flowchart: Connector [id:dp9434166458849813] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (185.88,85.87) .. controls (185.88,84.15) and (187.05,82.75) .. (188.5,82.75) .. controls (189.95,82.75) and (191.13,84.15) .. (191.13,85.88) .. controls (191.13,87.6) and (189.95,89) .. (188.5,89) .. controls (187.05,89) and (185.88,87.6) .. (185.88,85.87) -- cycle ;
%Flowchart: Connector [id:dp4009539143825589] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (185.88,62.87) .. controls (185.88,61.15) and (187.05,59.75) .. (188.5,59.75) .. controls (189.95,59.75) and (191.13,61.15) .. (191.13,62.88) .. controls (191.13,64.6) and (189.95,66) .. (188.5,66) .. controls (187.05,66) and (185.88,64.6) .. (185.88,62.87) -- cycle ;
%Flowchart: Connector [id:dp043881174465782546] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (109.88,62) .. controls (109.88,60.27) and (111.05,58.87) .. (112.5,58.88) .. controls (113.95,58.88) and (115.13,60.27) .. (115.12,62) .. controls (115.12,63.73) and (113.95,65.13) .. (112.5,65.12) .. controls (111.05,65.12) and (109.87,63.73) .. (109.88,62) -- cycle ;
%Straight Lines [id:da2794964806872138] 
\draw [color={rgb, 255:red, 103; green, 199; blue, 0 }  ,draw opacity=1 ]   (112.5,62) -- (189.5,62.88) ;
%Straight Lines [id:da40129325189267884] 
\draw [color={rgb, 255:red, 98; green, 189; blue, 7 }  ,draw opacity=1 ]   (112.5,62) -- (112.5,123.38) ;
%Straight Lines [id:da8622697956168519] 
\draw [color={rgb, 255:red, 85; green, 160; blue, 0 }  ,draw opacity=1 ]   (188.5,62.88) -- (188.5,85.88) ;
%Straight Lines [id:da033013833661442504] 
\draw [color={rgb, 255:red, 92; green, 170; blue, 0 }  ,draw opacity=1 ]   (111.5,126.5) -- (188.5,85.88) ;
%Straight Lines [id:da5130165638526303] 
\draw [color={rgb, 255:red, 245; green, 166; blue, 35 }  ,draw opacity=1 ]   (219,40) -- (221,195) ;
%Flowchart: Connector [id:dp39949874189366485] 
\draw  [color={rgb, 255:red, 245; green, 166; blue, 35 }  ,draw opacity=1 ] (253,119.5) .. controls (253,88.3) and (278.52,63) .. (310,63) .. controls (341.48,63) and (367,88.3) .. (367,119.5) .. controls (367,150.7) and (341.48,176) .. (310,176) .. controls (278.52,176) and (253,150.7) .. (253,119.5) -- cycle ;
%Flowchart: Connector [id:dp8703601517357209] 
\draw  [color={rgb, 255:red, 248; green, 15; blue, 212 }  ,draw opacity=1 ] (362,147.5) .. controls (362,134.52) and (372.07,124) .. (384.5,124) .. controls (396.93,124) and (407,134.52) .. (407,147.5) .. controls (407,160.48) and (396.93,171) .. (384.5,171) .. controls (372.07,171) and (362,160.48) .. (362,147.5) -- cycle ;
%Straight Lines [id:da6650140504044444] 
\draw [color={rgb, 255:red, 36; green, 111; blue, 240 }  ,draw opacity=1 ]   (359,88) -- (399,131) ;
%Flowchart: Connector [id:dp3619164566631141] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (355.25,88) .. controls (355.25,85.65) and (356.93,83.75) .. (359,83.75) .. controls (361.07,83.75) and (362.75,85.65) .. (362.75,88) .. controls (362.75,90.35) and (361.07,92.25) .. (359,92.25) .. controls (356.93,92.25) and (355.25,90.35) .. (355.25,88) -- cycle ;
%Flowchart: Connector [id:dp9347263069659886] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (395.25,131) .. controls (395.25,128.65) and (396.93,126.75) .. (399,126.75) .. controls (401.07,126.75) and (402.75,128.65) .. (402.75,131) .. controls (402.75,133.35) and (401.07,135.25) .. (399,135.25) .. controls (396.93,135.25) and (395.25,133.35) .. (395.25,131) -- cycle ;
%Flowchart: Connector [id:dp2845437783922642] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (380.38,147.5) .. controls (380.38,145.01) and (382.22,143) .. (384.5,143) .. controls (386.78,143) and (388.63,145.01) .. (388.63,147.5) .. controls (388.63,149.99) and (386.78,152) .. (384.5,152) .. controls (382.22,152) and (380.38,149.99) .. (380.38,147.5) -- cycle ;
%Flowchart: Connector [id:dp21467877467843643] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (305.88,119.5) .. controls (305.88,117.01) and (307.72,115) .. (310,115) .. controls (312.28,115) and (314.13,117.01) .. (314.13,119.5) .. controls (314.13,121.99) and (312.28,124) .. (310,124) .. controls (307.72,124) and (305.88,121.99) .. (305.88,119.5) -- cycle ;
%Straight Lines [id:da37855809127262074] 
\draw [color={rgb, 255:red, 245; green, 166; blue, 35 }  ,draw opacity=1 ]   (359,88) -- (310,119.5) ;
%Straight Lines [id:da020782590264599277] 
\draw [color={rgb, 255:red, 236; green, 17; blue, 178 }  ,draw opacity=1 ]   (399,131) -- (380.75,147.5) ;
%Straight Lines [id:da49270978044100855] 
\draw [color={rgb, 255:red, 245; green, 166; blue, 35 }  ,draw opacity=1 ]   (421,33) -- (423,188) ;
%Shape: Circle [id:dp7329040434972465] 
\draw  [color={rgb, 255:red, 4; green, 8; blue, 204 }  ,draw opacity=1 ] (439,120) .. controls (439,90.18) and (463.18,66) .. (493,66) .. controls (522.82,66) and (547,90.18) .. (547,120) .. controls (547,149.82) and (522.82,174) .. (493,174) .. controls (463.18,174) and (439,149.82) .. (439,120) -- cycle ;
%Flowchart: Connector [id:dp33620786872050124] 
\draw  [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ] (542,90.88) .. controls (542,78.17) and (552.07,67.88) .. (564.5,67.88) .. controls (576.93,67.88) and (587,78.17) .. (587,90.88) .. controls (587,103.58) and (576.93,113.88) .. (564.5,113.88) .. controls (552.07,113.88) and (542,103.58) .. (542,90.88) -- cycle ;
%Straight Lines [id:da6322795399987078] 
\draw [color={rgb, 255:red, 6; green, 160; blue, 47 }  ,draw opacity=1 ]   (564.5,67.88) -- (493,66) ;
%Straight Lines [id:da7834678980865475] 
\draw [color={rgb, 255:red, 243; green, 2; blue, 7 }  ,draw opacity=1 ]   (564.5,90.88) -- (587,90.88) ;
%Straight Lines [id:da742080265912703] 
\draw [color={rgb, 255:red, 5; green, 20; blue, 175 }  ,draw opacity=1 ]   (493,120) -- (536,152) ;
%Flowchart: Connector [id:dp562143748987187] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (489.25,120) .. controls (489.25,117.65) and (490.93,115.75) .. (493,115.75) .. controls (495.07,115.75) and (496.75,117.65) .. (496.75,120) .. controls (496.75,122.35) and (495.07,124.25) .. (493,124.25) .. controls (490.93,124.25) and (489.25,122.35) .. (489.25,120) -- cycle ;
%Flowchart: Connector [id:dp985537838160158] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (560.75,90.88) .. controls (560.75,88.53) and (562.43,86.63) .. (564.5,86.63) .. controls (566.57,86.63) and (568.25,88.53) .. (568.25,90.88) .. controls (568.25,93.22) and (566.57,95.13) .. (564.5,95.13) .. controls (562.43,95.13) and (560.75,93.22) .. (560.75,90.88) -- cycle ;
%Flowchart: Connector [id:dp21489204129579398] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (489.25,66) .. controls (489.25,63.65) and (490.93,61.75) .. (493,61.75) .. controls (495.07,61.75) and (496.75,63.65) .. (496.75,66) .. controls (496.75,68.35) and (495.07,70.25) .. (493,70.25) .. controls (490.93,70.25) and (489.25,68.35) .. (489.25,66) -- cycle ;
%Flowchart: Connector [id:dp7680178756330112] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (560.75,67.88) .. controls (560.75,65.53) and (562.43,63.63) .. (564.5,63.63) .. controls (566.57,63.63) and (568.25,65.53) .. (568.25,67.88) .. controls (568.25,70.22) and (566.57,72.13) .. (564.5,72.13) .. controls (562.43,72.13) and (560.75,70.22) .. (560.75,67.88) -- cycle ;

% Text Node
\draw (98,133) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{O}};
% Text Node
\draw (193.13,88.88) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{P}};
% Text Node
\draw (182,37) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{B}};
% Text Node
\draw (104,37) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{A}};
% Text Node
\draw (29,53) node [anchor=north west][inner sep=0.75pt]   [align=left] {1)};
% Text Node
\draw (231,51) node [anchor=north west][inner sep=0.75pt]   [align=left] {2)};
% Text Node
\draw (485,43) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{A}};
% Text Node
\draw (563,45) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{B}};
% Text Node
\draw (550,93) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{O}};
% Text Node
\draw (478.13,101.88) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{P}};
% Text Node
\draw (566.5,75.13) node [anchor=north west][inner sep=0.75pt]   [align=left] {4};
% Text Node
\draw (516,120) node [anchor=north west][inner sep=0.75pt]   [align=left] {9};
% Text Node
\draw (391.88,142.25) node [anchor=north west][inner sep=0.75pt]   [align=left] {9};
% Text Node
\draw (313,87) node [anchor=north west][inner sep=0.75pt]   [align=left] {25};
% Text Node
\draw (98,84) node [anchor=north west][inner sep=0.75pt]   [align=left] {8};
% Text Node
\draw (193.13,65.88) node [anchor=north west][inner sep=0.75pt]   [align=left] {2};
% Text Node
\draw (292,125) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{O}};
% Text Node
\draw (361,59) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{A}};
% Text Node
\draw (395,104) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{B}};
% Text Node
\draw (372.13,146.88) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{P}};

\end{tikzpicture}

Gimli · Accepted Answer

\begin{enumerate}
\item Введём в рисунок новые обозначения:
\begin{figure}[H]\centering

ikzset{every picture/.style={line width=1.25pt}} %set default line width to 0.75pt

\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,300); %set diagram left start at 0, and has height of 300

%Shape: Circle [id:dp8850246455431798] 
\draw  [color={rgb, 255:red, 189; green, 16; blue, 224 }  ,draw opacity=1 ] (60,110.5) .. controls (60,66.59) and (95.59,31) .. (139.5,31) .. controls (183.41,31) and (219,66.59) .. (219,110.5) .. controls (219,154.41) and (183.41,190) .. (139.5,190) .. controls (95.59,190) and (60,154.41) .. (60,110.5) -- cycle ;
%Shape: Circle [id:dp02794422110410344] 
\draw  [color={rgb, 255:red, 189; green, 16; blue, 224 }  ,draw opacity=1 ] (203,55) .. controls (203,41.19) and (214.19,30) .. (228,30) .. controls (241.81,30) and (253,41.19) .. (253,55) .. controls (253,68.81) and (241.81,80) .. (228,80) .. controls (214.19,80) and (203,68.81) .. (203,55) -- cycle ;
%Straight Lines [id:da9243719071028182] 
\draw [color={rgb, 255:red, 65; green, 117; blue, 5 }  ,draw opacity=1 ]   (139.5,31) -- (228,30) ;
%Straight Lines [id:da9450733938549347] 
\draw [color={rgb, 255:red, 65; green, 117; blue, 5 }  ,draw opacity=1 ]   (139.5,31) -- (139.5,110.5) ;
%Straight Lines [id:da04123768725136445] 
\draw [color={rgb, 255:red, 65; green, 117; blue, 5 }  ,draw opacity=1 ]   (228,30) -- (228,55) ;
%Straight Lines [id:da9894360692911766] 
\draw [color={rgb, 255:red, 65; green, 117; blue, 5 }  ,draw opacity=1 ]   (139.5,110.5) -- (228,55) ;
%Shape: Circle [id:dp19093648473979896] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (224,55) .. controls (224,52.79) and (225.79,51) .. (228,51) .. controls (230.21,51) and (232,52.79) .. (232,55) .. controls (232,57.21) and (230.21,59) .. (228,59) .. controls (225.79,59) and (224,57.21) .. (224,55) -- cycle ;
%Shape: Circle [id:dp27843613399582545] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (135.5,110.5) .. controls (135.5,108.29) and (137.29,106.5) .. (139.5,106.5) .. controls (141.71,106.5) and (143.5,108.29) .. (143.5,110.5) .. controls (143.5,112.71) and (141.71,114.5) .. (139.5,114.5) .. controls (137.29,114.5) and (135.5,112.71) .. (135.5,110.5) -- cycle ;
%Shape: Circle [id:dp22176142778940466] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (224,30) .. controls (224,27.79) and (225.79,26) .. (228,26) .. controls (230.21,26) and (232,27.79) .. (232,30) .. controls (232,32.21) and (230.21,34) .. (228,34) .. controls (225.79,34) and (224,32.21) .. (224,30) -- cycle ;
%Shape: Circle [id:dp5629652955476969] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (135.5,31) .. controls (135.5,28.79) and (137.29,27) .. (139.5,27) .. controls (141.71,27) and (143.5,28.79) .. (143.5,31) .. controls (143.5,33.21) and (141.71,35) .. (139.5,35) .. controls (137.29,35) and (135.5,33.21) .. (135.5,31) -- cycle ;
%Straight Lines [id:da37764575602500217] 
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 }  ,draw opacity=1 ] [dash pattern={on 4.5pt off 4.5pt}]  (140,57) -- (157.08,57.31) -- (228,55) ;
%Shape: Circle [id:dp9745456892232065] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (201.5,69) .. controls (201.5,66.79) and (203.29,65) .. (205.5,65) .. controls (207.71,65) and (209.5,66.79) .. (209.5,69) .. controls (209.5,71.21) and (207.71,73) .. (205.5,73) .. controls (203.29,73) and (201.5,71.21) .. (201.5,69) -- cycle ;
%Straight Lines [id:da5612876180796311] 
\draw    (151,57) -- (151,71) -- (139.5,70.75) ;

% Text Node
\draw (131,119) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{O}};
% Text Node
\draw (133,7) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{A}};
% Text Node
\draw (221,5) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{B}};
% Text Node
\draw (236,52) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{P}};
% Text Node
\draw (122,51) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{H}};
% Text Node
\draw (230,33) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\small 2}};
% Text Node
\draw (129,36) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\small 2}};
% Text Node
\draw (127,76) node [anchor=north west][inner sep=0.75pt]   [align=left] {6};
% Text Node
\draw (207.5,72) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{K}};

\end{tikzpicture}

\end{figure}

Так как мы поставили точку Н так чтобы АН = ВР, имеем НО = 8 - 2 = 6. Так как $OP= OK + KP=r_1+r_2$, имеем $OP=8+2=10.$ Следовательно, по Теореме Пифагора, имеем:
$$HP=\sqrt{OP^2-HO^2}=\sqrt{10^2-6^2}=8.$$
Так как АВ = НР, имеем
ewline
	extbf{Ответ: АВ = 8.}
\item Введём в рисунок новые обозначения:
\begin{figure}[ht]\centering

ikzset{every picture/.style={line width=1.25pt}} %set default line width to 0.75pt

\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,300); %set diagram left start at 0, and has height of 300

%Shape: Circle [id:dp8850246455431798] 
\draw  [color={rgb, 255:red, 245; green, 166; blue, 35 }  ,draw opacity=1 ] (95.41,20.47) .. controls (133.23,-1.83) and (181.97,10.75) .. (204.27,48.57) .. controls (226.57,86.39) and (214,135.13) .. (176.18,157.43) .. controls (138.36,179.73) and (89.62,167.16) .. (67.31,129.34) .. controls (45.01,91.52) and (57.59,42.78) .. (95.41,20.47) -- cycle ;
%Shape: Circle [id:dp02794422110410344] 
\draw  [color={rgb, 255:red, 224; green, 16; blue, 190 }  ,draw opacity=1 ] (214.78,118.1) .. controls (227.75,110.45) and (244.47,114.76) .. (252.12,127.73) .. controls (259.77,140.71) and (255.46,157.43) .. (242.48,165.08) .. controls (229.51,172.73) and (212.79,168.42) .. (205.14,155.44) .. controls (197.49,142.47) and (201.8,125.75) .. (214.78,118.1) -- cycle ;
%Straight Lines [id:da9243719071028182] 
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 }  ,draw opacity=1 ]   (204.27,48.57) -- (250.09,124.29) ;
%Straight Lines [id:da9450733938549347] 
\draw [color={rgb, 255:red, 245; green, 166; blue, 35 }  ,draw opacity=1 ]   (204.27,48.57) -- (135.79,88.95) ;
%Straight Lines [id:da04123768725136445] 
\draw [color={rgb, 255:red, 245; green, 35; blue, 225 }  ,draw opacity=1 ]   (250.09,124.29) -- (228.55,136.99) ;
%Straight Lines [id:da9894360692911766] 
\draw [color={rgb, 255:red, 65; green, 117; blue, 5 }  ,draw opacity=1 ]   (135.79,88.95) -- (228.55,136.99) ;
%Shape: Circle [id:dp19093648473979896] 
\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (226.52,133.54) .. controls (228.43,132.42) and (230.88,133.05) .. (232,134.96) .. controls (233.12,136.86) and (232.49,139.31) .. (230.59,140.43) .. controls (228.68,141.56) and (226.23,140.92) .. (225.11,139.02) .. controls (223.99,137.12) and (224.62,134.67) .. (226.52,133.54) -- cycle ;
%Shape: Circle [id:dp27843613399582545] 
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% Text Node
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% Text Node
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% Text Node
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% Text Node
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% Text Node
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% Text Node
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% Text Node
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\end{figure}
Так как мы поставили точку Н так чтобы АН = ВР, имеем $HO = 25 - 9 = 16$. Так как $OP= OK + KP=r_1+r_2$, имеем $OP=25+9=34.$ Следовательно, по Теореме Пифагора, имеем:
$$HP=\sqrt{OP^2-HO^2}=\sqrt{34^2-16^2}=30.$$
Так как АВ = НР, имеем
ewline
	extbf{Ответ: АВ = 30.}

\item Введём в рисунок новые обозначения:
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\end{figure}
Так как мы поставили точку Н так чтобы АН = ВР, имеем НО = 9-4=5. Так как $OP= OK + KP=r_1+r_2$, имеем $OP=9+4=13.$ Следовательно, по Теореме Пифагора, имеем:
$$HP=\sqrt{OP^2-HO^2}=\sqrt{13^2-5^2}=12.$$
Так как АВ = НР, имеем
ewline
	extbf{Ответ: АВ = 12.}
\end{enumerate}

Задание 10, стр 42

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Задание 10, стр 42

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