Lines and planes: distances and angles

1103061408

Level: 
A
The rectangular box \( ABCDEFGH \) has sides of lengths \( |AB|=|BC|=6\,\mathrm{cm} \), \( |AE|=8\,\mathrm{cm} \). Find the angle between the planes \( ABC \) and \( AFH \) (see the picture). Round the result to two decimal places.
\( 62.06^{\circ} \)
\( 53.13^{\circ} \)
\( 45^{\circ} \)
\( 60^{\circ} \)

2010015605

Level: 
A
The rectangular box \( ABCDA'B'C'D' \) has edges of lengths \( |AB|=6\,\mathrm{cm} \) and \( |BC|=8\,\mathrm{cm} \). The point \(S\) is the center of the base \(ABCD\) (see the picture) and the length of the line segment \(A'S\) is \(13\,\mathrm{cm}\). Find the distance between the points \(A\) and \(A'\).
\( 12\,\mathrm{cm} \)
\( \sqrt{194}\,\mathrm{cm} \)
\( \sqrt{69}\,\mathrm{cm} \)
\( 4\sqrt{10}\,\mathrm{cm} \)

2010015606

Level: 
A
The rectangular box \( ABCDA'B'C'D' \) has edges of lengths \( |AB|=4\sqrt3\,\mathrm{cm} \) and \( |BC|=8\,\mathrm{cm} \). The point \(S\) is the center of the lateral face \(ADD'A'\) (see the picture) and the length of the line segment \(B'S\) is \(10\,\mathrm{cm}\). Find the distance between the points \(A\) and \(A'\).
\( 12\,\mathrm{cm} \)
\( 6\,\mathrm{cm} \)
\( \sqrt{164}\,\mathrm{cm} \)
\( \sqrt{272}\,\mathrm{cm} \)

2010015607

Level: 
A
The rectangular box \( ABCDA'B'C'D' \) has edges of lengths \( |AB|=5\,\mathrm{cm} \) and \( |BC|=6\,\mathrm{cm} \). The distance between the center of the top face \(A'B'C'D'\) and the center of the bottom face \(ABCD\) is \(12\,\mathrm{cm}\). Find the length of the diagonal \(DC'\).
\( 13\,\mathrm{cm} \)
\( 6\sqrt5 \,\mathrm{cm} \)
\( \sqrt{119}\,\mathrm{cm} \)
\(6 \sqrt{3}\,\mathrm{cm} \)

2010015805

Level: 
A
A cuboid has sides \(a = 6\, \mathrm{cm}\) and \(b = 8\, \mathrm{cm}\), and the space diagonal \(u = 11\, \mathrm{cm}\). Find the length of the side \(c\) (see the picture).
\( \sqrt{21}\,\mathrm{cm} \)
\( \sqrt{221}\,\mathrm{cm} \)
\( 21\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)

2010015807

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 shortest face diagonal is \(u_{s}\). Find the ratio \(u_{s} : u_{t}\).
\(5 : 13\)
\(13 : 5\)
\(13\sqrt{10}:40\)
\(4\sqrt{10}:13\)

9000045709

Level: 
A
Let \(\omega \) be the angle between the solid diagonal of a box and the base of this box. Find the expression which allows to find \(\omega \).
\(\mathop{\mathrm{tg}}\nolimits \omega = \frac{\sqrt{2}} {2} \)
\(\cos \omega = \frac{\sqrt{2}} {2} \)
\(\sin \omega = \frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits \omega = \frac{\sqrt{2}} {2} \)

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).
\(10\sqrt{2}\, \mathrm{cm}\)
\(\sqrt{222}\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(2\sqrt{10}\, \mathrm{cm}\)
\(5\sqrt{7}\, \mathrm{cm}\)

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.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{\sqrt{2}} {2} \)
\(\sin \alpha = \frac{\sqrt{3}} {2} \)
\(\cos \alpha = \frac{\sqrt{5}} {3} \)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \sqrt{3}\)
\(\alpha = 45^{\circ }\)