Площадь трапеции равна произведению полусуммы ее оснований (a, b) на высоту (h)
(5+6):2=5.5
5.5*7=38.5 см(квадратных)
1.
![\frac{x+2}{3+2}= \frac{y+1}{1+1}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%2B2%7D%7B3%2B2%7D%3D+%5Cfrac%7By%2B1%7D%7B1%2B1%7D++)
![\frac{x+2}{5}= \frac{y+1}{2}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%2B2%7D%7B5%7D%3D+%5Cfrac%7By%2B1%7D%7B2%7D++)
2. c = 0,5m + n = (3 ; -1) + (1; -2) = (4; -3)
3. По теореме синусов:
![\frac{x}{sin45^0} = \frac{10}{sin60^0}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%7D%7Bsin45%5E0%7D+%3D+%5Cfrac%7B10%7D%7Bsin60%5E0%7D+)
![\frac{x}{ \frac{ \sqrt{2} }{2} } = \frac{10}{\frac{ \sqrt{3} }{2}}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%7D%7B+%5Cfrac%7B+%5Csqrt%7B2%7D+%7D%7B2%7D+%7D++%3D+%5Cfrac%7B10%7D%7B%5Cfrac%7B+%5Csqrt%7B3%7D+%7D%7B2%7D%7D)
![x = \frac{10 \sqrt{2} }{\sqrt{3}}=\frac{10 \sqrt{6} }{3}](https://tex.z-dn.net/?f=+x++%3D+%5Cfrac%7B10+%5Csqrt%7B2%7D+%7D%7B%5Csqrt%7B3%7D%7D%3D%5Cfrac%7B10+%5Csqrt%7B6%7D+%7D%7B3%7D)
см.
4. Обозначим АВ = 10 см, ВС = 8 см.
cos ∠B = cos 60° =
![\frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D)
sin ∠B = sin 60° =
![\frac{\sqrt{3}}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D)
По теореме косинусов:
AC² = AB² + BC² - 2·AB·BC·cos∠B
AC² = 10² + 8² - 2·10·8·0,5 = 100 + 64 - 80 = 84 см².
AC = 2√21 см
BC² = AB² + AC² - 2·AB·AC·cos∠A
Откуда: cos∠A =
![\frac{AB^2+AC^2-BC^2}{2*AB*AC}](https://tex.z-dn.net/?f=+%5Cfrac%7BAB%5E2%2BAC%5E2-BC%5E2%7D%7B2%2AAB%2AAC%7D+)
cos∠A =
![\frac{100+84-64}{2*10*2 \sqrt{21} }=\frac{120}{40\sqrt{21}}=\frac{3}{\sqrt{21}}=\frac{\sqrt{21}}{7}](https://tex.z-dn.net/?f=%5Cfrac%7B100%2B84-64%7D%7B2%2A10%2A2+%5Csqrt%7B21%7D+%7D%3D%5Cfrac%7B120%7D%7B40%5Csqrt%7B21%7D%7D%3D%5Cfrac%7B3%7D%7B%5Csqrt%7B21%7D%7D%3D%5Cfrac%7B%5Csqrt%7B21%7D%7D%7B7%7D)
AB² = BC² + AC² - 2·BC·AC·cos∠C
Откуда: cos∠C =
![\frac{BC^2+AC^2-AB^2}{2*BC*AC}](https://tex.z-dn.net/?f=+%5Cfrac%7BBC%5E2%2BAC%5E2-AB%5E2%7D%7B2%2ABC%2AAC%7D+)
cos∠C =
![\frac{64+84-100}{2*8*2\sqrt{21}}=\frac{48}{32\sqrt{21}}=\frac{3}{2\sqrt{21}}=\frac{\sqrt{21}}{14}](https://tex.z-dn.net/?f=%5Cfrac%7B64%2B84-100%7D%7B2%2A8%2A2%5Csqrt%7B21%7D%7D%3D%5Cfrac%7B48%7D%7B32%5Csqrt%7B21%7D%7D%3D%5Cfrac%7B3%7D%7B2%5Csqrt%7B21%7D%7D%3D%5Cfrac%7B%5Csqrt%7B21%7D%7D%7B14%7D)
Поскольку cos∠A и cos∠C -- положительные, ∠A и ∠C -- острые.
Следовательно, их синусы тоже положительные:
![sin\ \textless \ A=\sqrt{1-cos^2\ \textless \ A}](https://tex.z-dn.net/?f=+sin%5C+%5Ctextless+%5C+A%3D%5Csqrt%7B1-cos%5E2%5C+%5Ctextless+%5C+A%7D+)
![sin\ \textless \ A=\sqrt{1- ({\frac{ \sqrt{21} }{7})^2}}=\sqrt{1-{\frac{21}{49}}}=\sqrt{\frac{28}{49}}=\frac{\sqrt{28}}{7}=\frac{2\sqrt{7}}{7}](https://tex.z-dn.net/?f=sin%5C+%5Ctextless+%5C+A%3D%5Csqrt%7B1-+%28%7B%5Cfrac%7B+%5Csqrt%7B21%7D+%7D%7B7%7D%29%5E2%7D%7D%3D%5Csqrt%7B1-%7B%5Cfrac%7B21%7D%7B49%7D%7D%7D%3D%5Csqrt%7B%5Cfrac%7B28%7D%7B49%7D%7D%3D%5Cfrac%7B%5Csqrt%7B28%7D%7D%7B7%7D%3D%5Cfrac%7B2%5Csqrt%7B7%7D%7D%7B7%7D)
![sin\ \textless \ C=\sqrt{1-cos^2\ \textless \ C}](https://tex.z-dn.net/?f=+sin%5C+%5Ctextless+%5C+C%3D%5Csqrt%7B1-cos%5E2%5C+%5Ctextless+%5C+C%7D+)
![sin\ \textless \ C=\sqrt{1- ({\frac{ \sqrt{21} }{14})^2}}=\sqrt{1-{\frac{21}{196}}}=\sqrt{\frac{175}{196}}=\frac{\sqrt{175}}{14}=\frac{5\sqrt{7}}{14}](https://tex.z-dn.net/?f=sin%5C+%5Ctextless+%5C+C%3D%5Csqrt%7B1-+%28%7B%5Cfrac%7B+%5Csqrt%7B21%7D+%7D%7B14%7D%29%5E2%7D%7D%3D%5Csqrt%7B1-%7B%5Cfrac%7B21%7D%7B196%7D%7D%7D%3D%5Csqrt%7B%5Cfrac%7B175%7D%7B196%7D%7D%3D%5Cfrac%7B%5Csqrt%7B175%7D%7D%7B14%7D%3D%5Cfrac%7B5%5Csqrt%7B7%7D%7D%7B14%7D)