Metric properties

9000120305

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) shown in the picture is \(a = 3\, \mathrm{cm}\) and the height is \(v = 8\, \mathrm{cm}\). Find the angle between the diagonal \(AD'\) and the base plane \(ABC\) (round your result to the nearest degree).
\(53^{\circ }\)
\(37^{\circ }\)
\(45^{\circ }\)
\(61^{\circ }\)
\(72^{\circ }\)

9000120309

Level: 
A
The sides of a rectangular box shown in the picture are \(a = 3\, \mathrm{cm}\), \(b = 4\, \mathrm{cm}\), and \(c = 12\, \mathrm{cm}\). The space diagonal is \(u_{t}\) and the longest face diagonal is \(u_{s}\). Find the ratio \(u_{t} : u_{s}\).
\(13\sqrt{10} : 40\)
\(13 : \sqrt{153}\)
\(13 : 12\)
\(4\sqrt{10} : 5\)
\(4\sqrt{10} : 13\)

9000120304

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) is \(a = 3\, \mathrm{cm}\) and the height \(v = 8\, \mathrm{cm}\). Find the length of the diagonal \(AD'\).
\(10\, \mathrm{cm}\)
\(\sqrt{73}\, \mathrm{cm}\)
\(\sqrt{82}\, \mathrm{cm}\)
\(2\sqrt{8}\, \mathrm{cm}\)
\(2\sqrt{6}\, \mathrm{cm}\)