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МатематикаГДЗ по геометрии 8 класс Атанасян Задание 499
Задание 499
\[\boxed{\mathbf{499.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\ задачи:\]
\[\mathbf{Дано:}\]
\[\mathrm{\Delta}\text{ABC};\]
\[\textbf{а)}\ \text{AB} = 24\ см;\]
\[\text{BC} = 25\ см;\]
\[\text{AC} = 7\ см.\]
\[\textbf{б)}\ \text{AB} = 15\ см;\]
\[\text{BC} = 17\ см;\]
\[\text{AC} = 8\ см\text{.\ }\]
\[\mathbf{Найти:}\]
\[наименьшую\ высоту.\]
\[\mathbf{Решение.}\]
\[\textbf{а)}\ 1)\ По\ формуле\ Герона\ \]
\[\left( \ p = \frac{a + b + c}{2} \right):\]
\[S_{\text{ABC}} = \sqrt{p(p - a)(p - b)(p - c)};\]
\[p = \frac{24 + 25 + 7}{2} = 28\ см.\]
\[2)\ S_{\text{ABC}} =\]
\[= \sqrt{28(28 - 24)(28 - 25)(28 - 7)} =\]
\[= \sqrt{28 \bullet 4 \bullet 3 \bullet 21} =\]
\[= \sqrt{7 \bullet 4 \bullet 4 \bullet 3 \bullet 3 \bullet 7} = 3 \bullet 4 \bullet 7 =\]
\[= 84\ см^{2}.\]
\[3)\ S_{\text{ABC}} = \frac{1}{2} \bullet \text{BH} \bullet \text{AC} = 84\]
\[\frac{1}{2}\text{BH} \bullet 7 = 84;\]
\[\text{BH} = \frac{84 \bullet 2}{7} = 24\ см.\]
\[S_{\text{ABC}} = \frac{1}{2} \bullet \text{CF} \bullet \text{AB} = 84\]
\[\frac{1}{2}\text{CF} \bullet 24 = 84\]
\[\text{CF} = \frac{84 \bullet 2}{24} = 7\ см.\]
\[S_{\text{ABC}} = \frac{1}{2} \bullet \text{AE} \bullet \text{BC} = 84\]
\[\frac{1}{2}\text{AE} \bullet 25 = 84\]
\[\text{AE} = \frac{84 \bullet 2}{25} = \frac{168}{25} = 6,72\ см.\]
\[\mathbf{Ответ}:\text{AE} = 6,72\ см.\]
\[\textbf{б)}\ 1)\ По\ формуле\ Герона\ \]
\[\left( = \frac{a + b + c}{2} \right):\]
\[S_{\text{ABC}} = \sqrt{p(p - a)(p - b)(p - c)};\]
\[p = \frac{15 + 17 + 8}{2} = 20\ см.\]
\[2)\ S_{\text{ABC}} =\]
\[= \sqrt{20(20 - 15)(20 - 17)(20 - 8)} =\]
\[= \sqrt{20 \bullet 5 \bullet 3 \bullet 12} =\]
\[= \sqrt{4 \bullet 5 \bullet 5 \bullet 3 \bullet 3 \bullet 4} = 5 \bullet 4 \bullet 3 =\]
\[= 60\ см^{2}.\]
\[3)\ S_{\text{ABC}} = \frac{1}{2} \bullet \text{BH} \bullet \text{AC} = 60\]
\[\frac{1}{2}\text{BH} \bullet 8 = 60\]
\[\text{BH} = \frac{60 \bullet 2}{8} = 15\ см.\]
\[S_{\text{ABC}} = \frac{1}{2} \bullet \text{CF} \bullet \text{AB} = 60\]
\[\frac{1}{2}\text{CF} \bullet 15 = 60\]
\[\text{CF} = \frac{60 \bullet 2}{15} = 8\ см.\]
\[S_{\text{ABC}} = \frac{1}{2} \bullet \text{AE} \bullet \text{BC} = 60\]
\[\frac{1}{2}\text{AE} \bullet 17 = 60\]
\[\text{AE} = \frac{60 \bullet 2}{17} = \frac{120}{17} = 7\frac{1}{17}\ см.\]
\[\mathbf{Ответ}:\text{AE} = 7\frac{1}{17}\mathbf{.}\]
