Răspuns:
[tex]P_{\triangle MNP}=3\sqrt{3};\\A_{\triangle MNP}=\frac{3\sqrt{3}}{4}.[/tex]
Explicație pas cu pas:
[tex]\tan{\sphericalangle MNP}=\sqrt{3}\Rightarrow m\sphericalangle MNP=60\°\\\substack{MN\equiv MP}\Rightarrow \triangle MNP \text{ isoscel}.\\\\\substack{\triangle MNP\text{ isoscel}\\m\sphericalangle MNP=60\°}\}\Rightarrow\triangle MNP\text{ echilateral}[/tex]
Deoarece [tex]\triangle MNP[/tex] echi., putem folosi formulele acestui tip de triunghi:
[tex]P_{\triangle\text{echi.}}}=\text{latura}\cdot3[/tex];
[tex]A_{\triangle\text{echi.}}=\frac{\text{latura}^2\sqrt{3}}{4}[/tex].
Deci:
[tex]P_{\triangle MNP}=3\cdot MN\\\Rightarrow P_{\triangle MNP}=3\sqrt{3}.[/tex]
[tex]A_{\triangle MNP}=\frac{MN^2\sqrt{3}}{4}\\=\frac{\sqrt{3}^2\sqrt3}{4}\\=\frac{3\sqrt{3}}{4}[/tex]
