گام به گام تمرین صفحه 45 درس 2 ریاضی (2) (هندسه)

1)

الف)

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

ب)

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

پ)

\(\begin{array}{l}\left\{ \begin{array}{l}\widehat B = \widehat D = {90^ \circ }\\A\widehat CB = D\widehat CE\end{array} \right. \Rightarrow A\mathop C\limits^\Delta  B \sim D\mathop C\limits^\Delta  E\\ \Rightarrow \frac{{AB}}{{DE}} = \frac{{AC}}{{CE}} = \frac{{BC}}{{CD}} \Rightarrow \frac{4}{{12}} = \frac{{\frac{{20}}{3}}}{y} = \frac{x}{{16}} \Rightarrow \left\{ \begin{array}{l}x = \frac{{16}}{3}\\y = 20\end{array} \right.\end{array}\)

2)

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)

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

3)

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

4)

\(\begin{array}{l}\left\{ \begin{array}{l}\widehat A = \quad \quad \quad \\\widehat C = \widehat E = {90^ \circ }\end{array} \right. \Rightarrow A\mathop B\limits^\Delta  C \sim A\mathop D\limits^\Delta  E\\ \Rightarrow \frac{{AC}}{{AE}} = \frac{{BC}}{{DE}} \Rightarrow \frac{5}{{AE}} = \frac{1}{{60}} \Rightarrow AE = 300\\\\ \Rightarrow \frac{{AB}}{{DE}} = \frac{{AC}}{{CE}} = \frac{{BC}}{{CD}} \Rightarrow \frac{4}{{12}} = \frac{{\frac{{20}}{3}}}{y} = \frac{x}{{16}} \Rightarrow \left\{ \begin{array}{l}x = \frac{{16}}{3}\\y = 20\end{array} \right.\end{array}\)

5)

\(\begin{array}{l}\left\{ \begin{array}{l}\widehat B = \widehat D = {90^ \circ }\quad \quad \quad \\A\widehat CB = E\widehat CD\end{array} \right. \Rightarrow A\mathop B\limits^\Delta  C \sim C\mathop D\limits^\Delta  E\\\\ \Rightarrow \frac{{DC}}{{BC}} = \frac{{DE}}{{AB}} = \frac{{CE}}{{AC}} = \frac{{15}}{5} = 3\\\\\frac{{{P_{CDE}}}}{{{P_{ABC}}}} = \frac{{DC + DE + CE}}{{BC + AB + AC}} = \frac{{CE}}{{AC}} \Rightarrow \frac{{{P_{CDE}}}}{{{P_{ABC}}}} = 3\\\\\frac{{{S_{CDE}}}}{{{S_{ABC}}}} = \frac{{\frac{1}{2}DC \times DE}}{{\frac{1}{2}BC \times AB}} = \frac{{DC}}{{BC}} \times \frac{{DE}}{{AB}}\\ \Rightarrow \frac{{{S_{CDE}}}}{{{S_{ABC}}}} = 3 \times 3 \Rightarrow \frac{{{S_{CDE}}}}{{{S_{ABC}}}} = 9\end{array}\)

6)

الف)

\(\begin{array}{l}\left\{ \begin{array}{l}\widehat B = \widehat {B'}\quad \quad \quad \\\widehat {{H_1}} = \widehat {{{H'}_1}} = {90^ \circ }\end{array} \right. \Rightarrow A\mathop B\limits^\Delta  H \sim A'\mathop {B'}\limits^\Delta  H'\\\\ \Rightarrow \frac{{AB}}{{A'B'}} = \frac{{AH}}{{A'H'}} = \frac{{CH}}{{C'H'}} = k\end{array}\)

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)

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