جواب تمرین صفحه 45 درس 2 ریاضی یازدهم تجربی (هندسه)

الف

\(\begin{array}{l}\frac{{AB}}{{A'B'}} = \frac{{AC}}{{A'C'}} = \frac{{BC}}{{B'C'}}\\\\ \Rightarrow \frac{{2a}}{a} = \frac{{2b}}{b} = \frac{{2c}}{c} = 2\\\\ \Rightarrow A\mathop B\limits^\Delta C\sim A'\mathop {B'}\limits^\Delta C'\\\\ \Rightarrow x = \hat A' = \hat A\\\\ \Rightarrow x = {60^\circ } + {20^\circ } = {80^\circ }\end{array}\)

ب

\(\begin{array}{l}\left\{ {\begin{array}{*{20}{l}}\begin{array}{l}\frac{{AB}}{{DE}} = \frac{{AC}}{{DC}} = 2\\\end{array}\\{A\hat CB = D\hat CE}\end{array}} \right. \Rightarrow A\mathop C\limits^\Delta B\sim D\mathop C\limits^\Delta E\\\\ \Rightarrow \frac{{BC}}{{EC}} = \frac{{AB}}{{DE}}\\\\ \Rightarrow \frac{x}{{2/5}} = \frac{6}{3} = 2\\\\ \Rightarrow x = 7\end{array}\)

پ

\(\left\{ {\begin{array}{*{20}{l}}\begin{array}{l}\hat B = \hat D = {90^\circ }\\\end{array}\\{A\hat CB = D\hat CE}\end{array}} \right. \Rightarrow A\mathop C\limits^\Delta B\sim D\mathop C\limits^\Delta E\)

\(\begin{array}{l}\\ \Rightarrow \frac{{AB}}{{DE}} = \frac{{AC}}{{CE}} = \frac{{BC}}{{CD}}\\\\ \Rightarrow \frac{4}{{12}} = \frac{{\frac{{20}}{3}}}{y} = \frac{x}{{16}}\\\\ \Rightarrow \left\{ {\begin{array}{*{20}{l}}\begin{array}{l}x = \frac{{16}}{3}\\\end{array}\\{y = 20}\end{array}} \right.\end{array}\)

الف

\(\begin{array}{l}A{B^2} = BH \times BC = 9 \times 10 = 90\\\\ \Rightarrow AB = 3\sqrt {10} \\\\A{C^2} = B{C^2} - A{B^2} = 100 - 90 = 10\\\\ \Rightarrow AC = \sqrt {10} \\\\AB \times AC = AH \times BC\\\\ \Rightarrow 3\sqrt {10} \times \sqrt {10} = AH \times 10\\\\ \Rightarrow AH = 3\end{array}\)

ب

\(\begin{array}{l}A{C^2} = CH \times BC\\\\ \Rightarrow 25 = 2 \times BC\\\\ \Rightarrow BC = \frac{{25}}{2}\\\\A{B^2} = B{C^2} - A{C^2} = \frac{{625}}{4} - 25 = \frac{{525}}{4}\\\\ \Rightarrow AB = \frac{{5\sqrt {21} }}{2}\\\\AB \times AC = AH \times BC\\\\ \Rightarrow \frac{{5\sqrt {21} }}{2} \times 5 = AH \times \frac{{25}}{2}\\\\ \Rightarrow AH = \sqrt {21} \end{array}\)

پ

\(\begin{array}{l}B{C^2} = A{B^2} + A{C^2} = 64 + 36 = 100\\\\ \Rightarrow BC = 10\\\\AB \times AC = AH \times BC\\\\ \Rightarrow 8 \times 6 = AH \times 10\\\\ \Rightarrow AH = \frac{{48}}{{10}}\end{array}\)

ت

\(\begin{array}{l}B{H^2} = A{B^2} - A{H^2} = 144 - 36 = 108\\\\ \Rightarrow BH = 6\sqrt 3 \\\\A{B^2} = BH \times BC\\\\ \Rightarrow 144 = 6\sqrt 3 \times BC\\\\ \Rightarrow BC = 8\sqrt 3 \\\\A{C^2} = B{C^2} - A{B^2} = 192 - 144 = 48\\\\ \Rightarrow AC = 4\sqrt 3 \end{array}\)

\(\begin{array}{l}A{H^2} = A{B^2} - B{H^2} = 144 - 121 = 23\\\\ \Rightarrow AH = h = \sqrt {23} \\\\A{B^2} = BH \times BD\\\\ \Rightarrow 144 = 11 \times BD\\\\ \Rightarrow BD = \frac{{144}}{{11}}\\\\AD \times AB = AH \times BD\\\\ \Rightarrow x = \frac{{\sqrt {23} \times \frac{{144}}{{11}}}}{{12}}\\\\ \Rightarrow x = \frac{{12}}{{11}}\sqrt {23} \end{array}\)

\(\left\{ {\begin{array}{*{20}{l}}{\hat A = \hat A\quad \quad \quad }\\{\hat C = \hat E = {{90}^\circ }}\end{array}} \right. \Rightarrow A\mathop B\limits^\Delta C\sim A\mathop D\limits^\Delta E\)

\(\begin{array}{l}\\ \Rightarrow \frac{{AC}}{{AE}} = \frac{{BC}}{{DE}} \Rightarrow \frac{5}{{AE}} = \frac{1}{{60}}\\\\ \Rightarrow AE = 300\\\\ \Rightarrow \frac{{AB}}{{DE}} = \frac{{AC}}{{CE}} = \frac{{BC}}{{CD}}\\\\ \Rightarrow \frac{4}{{12}} = \frac{{\frac{{20}}{3}}}{y} = \frac{x}{{16}}\\\\ \Rightarrow \left\{ {\begin{array}{*{20}{l}}\begin{array}{l}x = \frac{{16}}{3}\\\end{array}\\{y = 20}\end{array}} \right.\end{array}\)

 \(\left\{ {\begin{array}{*{20}{l}}{\hat B = \hat D = {{90}^\circ }}\\{A\hat CB = E\hat CD}\end{array}} \right.\,\,\,\,\, \Rightarrow A\mathop B\limits^\Delta C\sim C\mathop D\limits^\Delta E\)

\(\begin{array}{l}\\ \Rightarrow \frac{{DC}}{{BC}} = \frac{{DE}}{{AB}} = \frac{{CE}}{{AC}} = \frac{{15}}{5} = 3\\\\\frac{{{P_{CDE}}}}{{{P_{ABC}}}} = \frac{{DC + DE + CE}}{{BC + AB + AC}} = \frac{{CE}}{{AC}}\\\\ \Rightarrow \frac{{{P_{CDE}}}}{{{P_{ABC}}}} = 3\\\\\frac{{{S_{CDE}}}}{{{S_{ABC}}}} = \frac{{\frac{1}{2}DC \times DE}}{{\frac{1}{2}BC \times AB}} = \frac{{DC}}{{BC}} \times \frac{{DE}}{{AB}}\\\\ \Rightarrow \frac{{{S_{CDE}}}}{{{S_{ABC}}}} = 3 \times 3\\\\ \Rightarrow \frac{{{S_{CDE}}}}{{{S_{ABC}}}} = 9\end{array}\)

الف

\(\begin{array}{l}\left\{ {\begin{array}{*{20}{l}}\begin{array}{l}\hat B = \hat B'\\\quad \quad \quad \end{array}\\{{{\hat H}_1} = {{\hat H'}_1} = {{90}^\circ }}\end{array}} \right. \Rightarrow A\mathop B\limits^\Delta H\sim A'\mathop {B'}\limits^\Delta H'\\\\ \Rightarrow \frac{{AB}}{{A'B'}} = \frac{{AH}}{{A'H'}} = \frac{{CH}}{{C'H'}} = k\end{array}\)

ب

\(\frac{{AH}}{{A'H'}} = k\)

پ

\(\begin{array}{l}\frac{{{S_{ABC}}}}{{{S_{A'B'C'}}}} = \frac{{\frac{1}{2}AH \times BC}}{{\frac{1}{2}A'H' \times B'C'}} = \\\\\frac{{AH}}{{A'H'}} \times \frac{{BC}}{{B'C'}} = k \times k\\\\ \Rightarrow \frac{{{S_{ABC}}}}{{{S_{A'B'C'}}}} = {k^2}\end{array}\)

ت

\(\begin{array}{l}\frac{{{P_{ABC}}}}{{{P_{A'B'C'}}}} = \frac{{BC + AB + AC}}{{B'C' + A'B' + A'C'}} = \frac{{AC}}{{A'C'}} = k\\\\ \Rightarrow \frac{{{P_{ABC}}}}{{{P_{A'B'C'}}}} = k\end{array}\)