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3 problems match “tangent ratio”. Clear

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1. Abscissa from tangent of terminal side ●○○○○ 10 ptsThe terminal side of angle $\alpha$ passes through the point $P(x, -6)$, and $\tan\alpha = -\frac{3}{5}$. Find the value of $x$.
2. Apothem of a regular pentagon ●●○○○ 20 ptsA regular pentagon has sides of length 4 m. Find the length of its apothem, in metres. Round to two decimal places.
3. Sum of squared tangents in a tetrahedron section ●●●●○ 100 ptsA regular tetrahedron is intersected by a plane that is perpendicular to one of its faces and meets that face along a line $e$. The intersection lines of this plane with the other three face planes of the tetrahedron form angles $\varphi_1, \varphi_2, \varphi_3$ with $e$. Determine the value of $\tan^2 \varphi_1+\tan^2 \varphi_2+\tan^2 \varphi_3$.