Trójkąt równoboczny ⇒ trójkąt, którego wszystkie boki maja taką samą długość.
a - długość boku trójkąta równobocznego
Korzystam ze wzorów:
[tex]Obw = a+a+a = 3a\\\\P_{\Delta rownobocznego} =\dfrac{a^{2} \sqrt{3} }{4} \\\\\\h_{\Delta rownobocznego} =\dfrac{a \sqrt{3} }{2}[/tex]
[tex]zad.1\\\\a)~~a=4~ [j]\\\\Obw =3\cdot 4 ~[j]= 12~ [j]\\\\P_{\Delta rownobocznego} =\dfrac{4^{2} \sqrt{3} }{4}~ [j^{2} ]=4\sqrt{3} ~[j^{2} ]\\\\\\b)~~a=\sqrt{3} ~ [j]\\\\Obw =3\cdot \sqrt{3} ~[j]= 3\sqrt{3} ~ [j]\\\\P_{\Delta rownobocznego} =\dfrac{(\sqrt{3} )^{2} \sqrt{3} }{4}~ [j^{2} ]=\dfrac{3\sqrt{3} }{4} ~~[j]^{2} ]\\\\\\[/tex]
[tex]c)~~a=5\sqrt{2} ~ [j]\\\\Obw =3\cdot 5\sqrt{2} ~[j]= 15\sqrt{2} ~ [j]\\\\P_{\Delta rownobocznego} =\dfrac{(5\sqrt{2} )^{2} \sqrt{3} }{4}~ [j]^{2} =\dfrac{50\sqrt{3} }{4} ~~[j^{2} ]=\dfrac{25\sqrt{3} }{2} ~~[j^{2} ]\\\\\\[/tex]
[tex]d)~~a=(\sqrt{2} +1)~ [j]\\\\Obw =3\cdot (\sqrt{2}+1) ~[j]=(3\sqrt{2} +3)~~[j]\\\\\ (\sqrt{2} +1)^{2} =(\sqrt{2} )^{2} +2\cdot \sqrt{2} \cdot 1+1^{2} =2+2\sqrt{2} +1=3+2\sqrt{2} \\\\\\\\P_{\Delta rownobocznego} =\dfrac{(\sqrt{2} +1 )^{2} \sqrt{3} }{4}~ [j]^{2} =\dfrac{(3+2\sqrt{2} )\cdot \sqrt{3} }{4} ~~[j^{2} ]=\dfrac{3\sqrt{3} +2\sqrt{6} }{4} ~~[j^{2} ][/tex]
[tex]zad.2\\\\a)~~h=4\sqrt{3}~[j] \\\\h=\dfrac{a\sqrt{3} }{2} ~~\land~~h=4\sqrt{3}~~\Rightarrow~~\dfrac{a\sqrt{3} }{2}=4\sqrt{3}\\\\\dfrac{a\sqrt{3} }{2}=4\sqrt{3}~~\mid \div \dfrac{\sqrt{3} }{2} \\\\a=8~[j]\\\\P_{\Delta rownobocznego} =\dfrac{a^{2} \sqrt{3} }{4} ~~\land~~a=8~[j]~~\Rightarrow~~P_{\Delta rownobocznego} =\dfrac{8^{2} \sqrt{3} }{4} ~[j^{2} ]\\\\P_{\Delta rownobocznego} =\dfrac{64 \sqrt{3} }{4} ~[j^{2} ]=16\sqrt{3} ~[j^{2} ]\\\\Obw=3\cdot 8 ~~[j]=24 ~~[j][/tex]
[tex]b)~~h=9~[j] \\\\h=\dfrac{a\sqrt{3} }{2} ~~\land~~h=9~~\Rightarrow~~\dfrac{a\sqrt{3} }{2}=9\\\\\dfrac{a\sqrt{3} }{2}=9~~\mid \div \dfrac{\sqrt{3} }{2} \\\\a=\dfrac{18}{\sqrt{3} } \cdot \dfrac{\sqrt{3} }{\sqrt{3} }=\dfrac{18\sqrt{3} }{3 } =6\sqrt{3}~[j] \\\\P_{\Delta rownobocznego} =\dfrac{a^{2} \sqrt{3} }{4} ~~\land~~a=6\sqrt{3} ~[j]~~\Rightarrow~~P_{\Delta rownobocznego} =\dfrac{(6\sqrt{3} )^{2} \sqrt{3} }{4} ~[j^{2} ]\\\\\\\\[/tex]
[tex]P_{\Delta rownobocznego} =\dfrac{36\cdot 3 \cdot \sqrt{3} }{4} ~[j^{2} ]\\\\P_{\Delta rownobocznego} =27\sqrt{3} ~~[j^{2} ]\\\\Obw=3\cdot 6\sqrt{3} ~~[j]=18\sqrt{3} ~~[j][/tex]
