1.stosunek długości przekątnych rombu jest równy 3:4. oblicz pole tego rombu, jeśli jego bok ma dłiugość 15 cm.
2.bok rombu ma długość 4 cm, a suma długości jego przekątnych jest równa 10 cm.oblicz pole i wysokość tego rombu.
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1
przekątne rombu
- bok rombu
![\begin{cases}\frac{e}{f}=\frac{3}{4} \\ (\frac{1}{2}e)^2+(\frac{1}{2}f )^2=a^2\end{cases} \begin{cases}\frac{e}{f}=\frac{3}{4} \\ (\frac{1}{2}e)^2+(\frac{1}{2}f )^2=a^2\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%5Cfrac%7Be%7D%7Bf%7D%3D%5Cfrac%7B3%7D%7B4%7D+%5C%5C+%28%5Cfrac%7B1%7D%7B2%7De%29%5E2%2B%28%5Cfrac%7B1%7D%7B2%7Df+%29%5E2%3Da%5E2%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ \frac{1}{4}e^2+\frac{1}{4}f^2=15^2\end{cases} \begin{cases}e=\frac{3}{4}f \\ \frac{1}{4}e^2+\frac{1}{4}f^2=15^2\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+%5Cfrac%7B1%7D%7B4%7De%5E2%2B%5Cfrac%7B1%7D%7B4%7Df%5E2%3D15%5E2%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ \frac{1}{4}\cdot (\frac{3}{4}f)^2+\frac{1}{4}f^2=225\end{cases} \begin{cases}e=\frac{3}{4}f \\ \frac{1}{4}\cdot (\frac{3}{4}f)^2+\frac{1}{4}f^2=225\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+%5Cfrac%7B1%7D%7B4%7D%5Ccdot+%28%5Cfrac%7B3%7D%7B4%7Df%29%5E2%2B%5Cfrac%7B1%7D%7B4%7Df%5E2%3D225%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ \frac{1}{4}\cdot \frac{9}{16}f^2+\frac{1}{4}f^2=225\end{cases} \begin{cases}e=\frac{3}{4}f \\ \frac{1}{4}\cdot \frac{9}{16}f^2+\frac{1}{4}f^2=225\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+%5Cfrac%7B1%7D%7B4%7D%5Ccdot+%5Cfrac%7B9%7D%7B16%7Df%5E2%2B%5Cfrac%7B1%7D%7B4%7Df%5E2%3D225%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ \frac{9}{64}f^2+\frac{1}{4}f^2=225 /\cdot 64\end{cases} \begin{cases}e=\frac{3}{4}f \\ \frac{9}{64}f^2+\frac{1}{4}f^2=225 /\cdot 64\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+%5Cfrac%7B9%7D%7B64%7Df%5E2%2B%5Cfrac%7B1%7D%7B4%7Df%5E2%3D225+%2F%5Ccdot+64%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ 9f^2+16f^2=14400\end{cases} \begin{cases}e=\frac{3}{4}f \\ 9f^2+16f^2=14400\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+9f%5E2%2B16f%5E2%3D14400%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ 25f^2=14400\ /:25\end{cases} \begin{cases}e=\frac{3}{4}f \\ 25f^2=14400\ /:25\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+25f%5E2%3D14400%5C+%2F%3A25%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ f^2=576\end{cases} \begin{cases}e=\frac{3}{4}f \\ f^2=576\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+f%5E2%3D576%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ f= \sqrt{576} \end{cases} \begin{cases}e=\frac{3}{4}f \\ f= \sqrt{576} \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+f%3D+%5Csqrt%7B576%7D+%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}f \\ f=24 \end{cases} \begin{cases}e=\frac{3}{4}f \\ f=24 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7Df+%5C%5C+f%3D24+%5Cend%7Bcases%7D)
![\begin{cases}e=\frac{3}{4}\cdot 24 \\ f=24 \end{cases} \begin{cases}e=\frac{3}{4}\cdot 24 \\ f=24 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D%5Cfrac%7B3%7D%7B4%7D%5Ccdot+24+%5C%5C+f%3D24+%5Cend%7Bcases%7D)
![\begin{cases}e=18 \\ f=24 \end{cases} \begin{cases}e=18 \\ f=24 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D18+%5C%5C+f%3D24+%5Cend%7Bcases%7D)
- przekątne rombu
- bok rombu
- wysokość rombu
![\begin{cases}e+f=10\\ (\frac{1}{2}e)^2+(\frac{1}{2}f )^2=a^2\end{cases} \begin{cases}e+f=10\\ (\frac{1}{2}e)^2+(\frac{1}{2}f )^2=a^2\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%2Bf%3D10%5C%5C+%28%5Cfrac%7B1%7D%7B2%7De%29%5E2%2B%28%5Cfrac%7B1%7D%7B2%7Df+%29%5E2%3Da%5E2%5Cend%7Bcases%7D)
![\begin{cases}e=10-f\\ \frac{1}{4}e^2+\frac{1}{4}f^2=4^2\end{cases} \begin{cases}e=10-f\\ \frac{1}{4}e^2+\frac{1}{4}f^2=4^2\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D10-f%5C%5C+%5Cfrac%7B1%7D%7B4%7De%5E2%2B%5Cfrac%7B1%7D%7B4%7Df%5E2%3D4%5E2%5Cend%7Bcases%7D)
![\begin{cases}e=10-f\\ \frac{1}{4}e^2+\frac{1}{4}f^2=16 \ /\cdot 4\end{cases} \begin{cases}e=10-f\\ \frac{1}{4}e^2+\frac{1}{4}f^2=16 \ /\cdot 4\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D10-f%5C%5C+%5Cfrac%7B1%7D%7B4%7De%5E2%2B%5Cfrac%7B1%7D%7B4%7Df%5E2%3D16+%5C+%2F%5Ccdot+4%5Cend%7Bcases%7D)
![\begin{cases}e=10-f\\ e^2+f^2=64\end{cases} \begin{cases}e=10-f\\ e^2+f^2=64\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D10-f%5C%5C+e%5E2%2Bf%5E2%3D64%5Cend%7Bcases%7D)
![\begin{cases}e=10-f\\ (10-f)^2+f^2=64\end{cases} \begin{cases}e=10-f\\ (10-f)^2+f^2=64\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D10-f%5C%5C+%2810-f%29%5E2%2Bf%5E2%3D64%5Cend%7Bcases%7D)
![\begin{cases}e=10-f\\ 100-20f+f^2+f^2-64=0\end{cases} \begin{cases}e=10-f\\ 100-20f+f^2+f^2-64=0\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D10-f%5C%5C+100-20f%2Bf%5E2%2Bf%5E2-64%3D0%5Cend%7Bcases%7D)
![\begin{cases}e=10-f\\ 2f^2-20f+36=0\end{cases} \begin{cases}e=10-f\\ 2f^2-20f+36=0\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7De%3D10-f%5C%5C+2f%5E2-20f%2B36%3D0%5Cend%7Bcases%7D)
![2f^2-20f+36=0 \ /:2 2f^2-20f+36=0 \ /:2](https://tex.z-dn.net/?f=2f%5E2-20f%2B36%3D0+%5C+%2F%3A2)
![f^2-10f+18=0 f^2-10f+18=0](https://tex.z-dn.net/?f=f%5E2-10f%2B18%3D0)
![\Delta=(-10)^2-4\cdot 1\cdot 18=100-72=28 \Delta=(-10)^2-4\cdot 1\cdot 18=100-72=28](https://tex.z-dn.net/?f=%5CDelta%3D%28-10%29%5E2-4%5Ccdot+1%5Ccdot+18%3D100-72%3D28)
![\sqrt{\Delta} = \sqrt{28}=2 \sqrt{7} \sqrt{\Delta} = \sqrt{28}=2 \sqrt{7}](https://tex.z-dn.net/?f=%5Csqrt%7B%5CDelta%7D+%3D+%5Csqrt%7B28%7D%3D2+%5Csqrt%7B7%7D)
![f_1= \frac{10-2 \sqrt{7} }{2}=5-\sqrt{7} f_1= \frac{10-2 \sqrt{7} }{2}=5-\sqrt{7}](https://tex.z-dn.net/?f=f_1%3D+%5Cfrac%7B10-2+%5Csqrt%7B7%7D+%7D%7B2%7D%3D5-%5Csqrt%7B7%7D)
![f_2= \frac{10+2 \sqrt{7} }{2}=5+\sqrt{7} f_2= \frac{10+2 \sqrt{7} }{2}=5+\sqrt{7}](https://tex.z-dn.net/?f=f_2%3D+%5Cfrac%7B10%2B2+%5Csqrt%7B7%7D+%7D%7B2%7D%3D5%2B%5Csqrt%7B7%7D)
lub ![\begin{cases} e=10-f\\ f=5+\sqrt{7}\end{cases} \begin{cases} e=10-f\\ f=5+\sqrt{7}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D+e%3D10-f%5C%5C+f%3D5%2B%5Csqrt%7B7%7D%5Cend%7Bcases%7D)
lub ![\begin{cases} e=10-(5+\sqrt{7})\\ f=5+\sqrt{7}\end{cases} \begin{cases} e=10-(5+\sqrt{7})\\ f=5+\sqrt{7}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D+e%3D10-%285%2B%5Csqrt%7B7%7D%29%5C%5C+f%3D5%2B%5Csqrt%7B7%7D%5Cend%7Bcases%7D)
lub ![\begin{cases} e=10-5-\sqrt{7})\\ f=5+\sqrt{7}\end{cases} \begin{cases} e=10-5-\sqrt{7})\\ f=5+\sqrt{7}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D+e%3D10-5-%5Csqrt%7B7%7D%29%5C%5C+f%3D5%2B%5Csqrt%7B7%7D%5Cend%7Bcases%7D)
lub ![\begin{cases} e=5-\sqrt{7})\\ f=5+\sqrt{7}\end{cases} \begin{cases} e=5-\sqrt{7})\\ f=5+\sqrt{7}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D+e%3D5-%5Csqrt%7B7%7D%29%5C%5C+f%3D5%2B%5Csqrt%7B7%7D%5Cend%7Bcases%7D)
i ![(5-\sqrt{7})cm (5-\sqrt{7})cm](https://tex.z-dn.net/?f=%285-%5Csqrt%7B7%7D%29cm)
![ah=\frac{ef}{2} ah=\frac{ef}{2}](https://tex.z-dn.net/?f=ah%3D%5Cfrac%7Bef%7D%7B2%7D)
![4h=\frac{(5+\sqrt{7})(5-\sqrt{7})}{2} 4h=\frac{(5+\sqrt{7})(5-\sqrt{7})}{2}](https://tex.z-dn.net/?f=4h%3D%5Cfrac%7B%285%2B%5Csqrt%7B7%7D%29%285-%5Csqrt%7B7%7D%29%7D%7B2%7D)
![4h=\frac{25-7}{2} 4h=\frac{25-7}{2}](https://tex.z-dn.net/?f=4h%3D%5Cfrac%7B25-7%7D%7B2%7D)
![4h=\frac{18}{2} 4h=\frac{18}{2}](https://tex.z-dn.net/?f=4h%3D%5Cfrac%7B18%7D%7B2%7D)
![4h=9 \ /:4 4h=9 \ /:4](https://tex.z-dn.net/?f=4h%3D9+%5C+%2F%3A4)
![h= \frac{9}{4}cm h= \frac{9}{4}cm](https://tex.z-dn.net/?f=h%3D+%5Cfrac%7B9%7D%7B4%7Dcm)
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Obliczam przekątne rombu:
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Przekątne rombu są równe:
Obliczam wysokość rombu