Question

1)Se da un triunghi dreptunghic cu un unghi de 30 grade si perimetrul de 120 cm.Sa se afle lungimile laturilor triunghiului.


2)Perimetrul unui romb este egal cu a, iar raportul lungimilor diagonalelor este egal cu 3:4.Sa se afle lungimile diagonalelor rombului.

Answer 1

   
[tex]\displaystyle\\ \texttt{Problema 1), ~Fig. 1}\\\\ AC = BC \sin 30^o = BC \times \frac{1}{2}= \boxed{\frac{BC}{2}}\\\\ AB = BC\cos 30^o = BC\times \frac{\sqrt{3}}{2} =\boxed{\frac{BC\sqrt{3}}{2}}\\\\ P = AB+BC+AC= \frac{BC\sqrt{3}}{2}+BC+\frac{BC}{2}=\\\\ =BC\left(\frac{\sqrt{3}}{2}+1+\frac{1}{2}\right)=BC\left(\frac{\sqrt{3}}{2}+ \frac{2}{2} +\frac{1}{2}\right)=\\\\ =\boxed{BC\times \frac{3+\sqrt{3}}{2}} [/tex]


[tex]\displaystyle\\ P=120~cm\\\\ BC\times \frac{3+\sqrt{3}}{2}=120~cm\\\\ BC= \frac{~~~~120~~~~}{\dfrac{3+\sqrt{3}}{2}} =120\times \frac{2}{3+\sqrt{3}} = \\\\ =120\times \frac{2(3-\sqrt{3})}{9-3}= \frac{120\times2(3-\sqrt{3})}{6}=\\\\=20\times2(3-\sqrt{3})=\boxed{40(3-\sqrt{3})~cm}\\\\ AB=\frac{BC\sqrt{3}}{2}=\frac{40(3-\sqrt{3})\times\sqrt{3}}{2}=\\\\ =20(3\sqrt{3}-3)=\boxed{60(\sqrt{3}-1)~cm}\\\\ AC = \frac{40(3-\sqrt{3})}{2}=\boxed{20(3-\sqrt{3})}[/tex]



[tex]\displaystyle\\ \texttt{Problema 2), ~Fig. 2}\\\\ \frac{AC}{BD} = \frac{3}{4} \\\\ \frac{AO}{BO} = \frac{ \frac{AC}{2}}{ \frac{BD}{2}} =\frac{AC}{2}\times \frac{2}{BD} = \frac{AC}{BD} = \frac{3}{4} \\\\ \Longrightarrow \text{In }~\Delta AOB \text{reportul catetelor = 3/4.}\\\\ AO= 3k~~~si~~~BO = 4k\\\\ AB= \sqrt{AO^2+BO^2}= \sqrt{(3k)^2+(4k)^2}= \sqrt{9k^2+16k^2}=\\\\ =\sqrt{9k^2+16k^2}=\sqrt{25k^2}=5k [/tex]


[tex]\displaystyle\\ P=a\\\\ \Longrightarrow ~AB = \frac{P}{4} = \frac{a}{4} \\\\ 5k \cdots \cdots \cdots \frac{a}{4}~~~(AB)\\\\ 3k\cdots \cdots \cdots AO\\\\ 4k\cdots \cdots \cdots BO\\\\ AO = \frac{3k}{5k}\times \frac{a}{4} =\boxed{\frac{3a}{20}} \\\\ BO = \frac{4k}{5k}\times \frac{a}{4} =\boxed{\frac{4a}{20}} \\\\ \text{Calculam diagonalele:}\\\\ AC = 2AO = 2\times\frac{3a}{20}=\boxed{\frac{3a}{10}}\\\\ BD=2BO=2\times\frac{4a}{20}=\boxed{\frac{2a}{5}}[/tex]