Metric properties

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 \(AD'\).
\(10\, \mathrm{cm}\)
\(\sqrt{73}\, \mathrm{cm}\)
\(\sqrt{82}\, \mathrm{cm}\)
\(2\sqrt{8}\, \mathrm{cm}\)
\(2\sqrt{6}\, \mathrm{cm}\)

9000120303

Level: 
A
Identify a valid relation involving the angle \(\alpha \) defined as an angle between a solid diagonal and a face diagonal through the same vertex in a cube.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{\sqrt{2}} {2} \)
\(\sin \alpha = \frac{\sqrt{3}} {2} \)
\(\cos \alpha = \frac{\sqrt{5}} {3} \)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \sqrt{3}\)
\(\alpha = 45^{\circ }\)

9000121004

Level: 
A
In the cube \(ABCDEFGH\) find the angle between the lines \(S_{AE}S_{HC}\) and \(S_{HC}S_{BF}\), where \(S_{AE}\), \(S_{HC}\) and \(S_{BF}\) are the centers of the segments \(AE\), \(HC\) and \(BF\), respectively.
\(53.13^{\circ }\)
\(26.57^{\circ }\)
\(60^{\circ }\)
\(36.87^{\circ }\)

9000120302

Level: 
A
A cuboid has sides \(a = 5\, \mathrm{cm}\), \(b = 8\, \mathrm{cm}\), and \(c = \sqrt{111}\, \mathrm{cm}\). Find the length of the cuboid’s space diagonal \(u\) (see the picture).
\(10\sqrt{2}\, \mathrm{cm}\)
\(\sqrt{222}\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(2\sqrt{10}\, \mathrm{cm}\)
\(5\sqrt{7}\, \mathrm{cm}\)

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).
\(53^{\circ }\)
\(37^{\circ }\)
\(45^{\circ }\)
\(61^{\circ }\)
\(72^{\circ }\)