Ermittle den Streckungsfaktor $k\backslash \; k$

Bekannt ist:$\backslash quad$$\overline{BD}=4m;\overline{B^{\mathrm{\prime }}C}=6m;\overline{EZ}=\mathrm{1,2}m;\overline{BZ}=56m\backslash overline\{BD\}=4m;\backslash quad\; \backslash overline\{B\text{'}C\}=6m;\backslash quad\; \backslash overline\{EZ\}=1\{,\}2m;\backslash quad\; \backslash overline\{BZ\}=56m$

$k={\displaystyle \frac{\overline{A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}}}{\overline{AB}}}\backslash hspace\{50mm\}k=\backslash dfrac\{\backslash overline\{A\text{'}B\text{'}\}\}\{\backslash overline\{AB\}\}$

$\overline{A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}}=\overline{B^{\mathrm{\prime }}C}-\overline{A^{\mathrm{\prime }}C}=6m-\mathrm{1,2}m\overline{A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}}=\mathrm{4,8}m\backslash overline\{A\text{'}B\text{'}\}=\backslash overline\{B\text{'}C\}-\backslash overline\{A\text{'}C\}=6\backslash \; m-1\{,\}2\backslash \; m\backslash \backslash \backslash overline\{A\text{'}B\text{'}\}=4\{,\}8\backslash \; m$

$\overline{AB}=\overline{BD}-\overline{AD}=4m-\mathrm{1,2}m\overline{AB}=\mathrm{2,8}m\backslash \; \backslash overline\{AB\}=\backslash overline\{BD\}-\backslash overline\{AD\}=4\backslash \; m-1\{,\}2\backslash \; m\backslash \backslash \backslash \; \backslash overline\{AB\}=2\{,\}8\backslash \; m$

$k={\displaystyle \frac{\mathrm{4,8}m}{\mathrm{2,8}m}}\Rightarrow k={\displaystyle \frac{12}{7}}\backslash hspace\{38mm\}k=\backslash dfrac\{4\{,\}8\backslash \; m\}\{2\{,\}8\backslash \; m\}\; \backslash quad\backslash Rightarrow\backslash quad\; k=\backslash dfrac\{12\}\{7\}\backslash$oder$k\approx \mathrm{1,7}\backslash \; k\backslash approx1\{,\}7$

Der Streckungsfaktor $k\backslash \; k\backslash$beträgt ungefähr $\mathrm{1,7}\backslash \; 1\{,\}7$.

Berechne die Länge der Strecke $[B^{\mathrm{\prime }}B]inm:[B\text{'}B]\backslash \; in\backslash \; m:$

${\displaystyle \frac{56+\overline{B^{\mathrm{\prime }}B}}{56}}={\displaystyle \frac{\overline{A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}}}{\overline{AB}}}\Rightarrow {\displaystyle \frac{56+\overline{B^{\mathrm{\prime }}B}}{56}}={\displaystyle \frac{\mathrm{4,8}}{\mathrm{2,8}}}\Rightarrow \overline{B^{\mathrm{\prime }}B}={\displaystyle \frac{\mathrm{268,8}-\mathrm{156,8}}{\mathrm{2,8}}}\backslash dfrac\{56+\backslash overline\{B\text{'}B\}\}\{56\}=\backslash dfrac\{\backslash overline\{A\text{'}B\text{'}\}\}\{\backslash overline\{AB\}\}\backslash quad\backslash Rightarrow\backslash quad\backslash dfrac\{56+\backslash overline\{B\text{'}B\}\}\{56\}=\backslash dfrac\{4\{,\}8\}\{2\{,\}8\}\backslash quad\backslash Rightarrow\backslash quad\backslash overline\{B\text{'}B\}=\backslash dfrac\{268\{,\}8-156\{,\}8\}\{2\{,\}8\}$

$\overline{B^{\mathrm{\prime }}B}=40m\backslash hspace\{76mm\}\backslash overline\{B\text{'}B\}=40\backslash \; m$