Lines and planes: distances and angles

1103061404

Level: 
A
The rectangular box \( ABCDEFGH \) has sides of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \), \( |AE|=8\,\mathrm{cm} \). Find the angle between the lines \( HG \) and \( AH \) (see the picture). Round the result to two decimal places.
\( 90^{\circ} \)
\( 85^{\circ} \)
\( 56.15^{\circ} \)
\( 33.85^{\circ} \)

1103061403

Level: 
A
The rectangular box \( ABCDEFGH \) has sides of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \), \( |AE|=8\,\mathrm{cm} \). Find the angle between the lines \( CG \) and \( SC \), where \( S \) is the midpoint of the diagonal \( EG \) (see the picture). Round the result to two decimal places.
\( 24.26^{\circ} \)
\( 45^{\circ} \)
\( 35.24^{\circ} \)
\( 63.21^{\circ} \)

1103061402

Level: 
A
The rectangular box \( ABCDEFGH \) has sides of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \), \( |AE|=8\,\mathrm{cm} \). Find the angle between the lines \( AC \) and \( SC \), where \( S \) is the midpoint of the diagonal \( EG \) (see the picture). Round the result to two decimal places.
\( 65.74^{\circ} \)
\( 70.28^{\circ} \)
\( 75^{\circ} \)
\( 45^{\circ} \)

1103061401

Level: 
A
The rectangular box \( ABCDEFGH \) has sides of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \), \( |AE|=8\,\mathrm{cm} \). Find the angle between the plane \( EFB \) and the plane \( HGB \) (see the picture). Round the result to two decimal places.
\( 26.57^{\circ} \)
\( 45^{\circ} \)
\( 22.5^{\circ} \)
\( 31.43^{\circ} \)

1103101209

Level: 
C
Let \( ABCD \) be a regular tetrahedron with an edge length of \( 3\,\mathrm{cm} \). Find the height \( |DT| \) of the tetrahedron (see the picture).
\( \sqrt6\,\mathrm{cm} \)
\( \frac{3\sqrt3}2\,\mathrm{cm} \)
\( \frac{\sqrt3}2\,\mathrm{cm} \)
\( 3\sqrt2\,\mathrm{cm} \)

1103101208

Level: 
C
Let \( ABCD \) be a regular tetrahedron with an edge length of \( 5\,\mathrm{cm} \). Let \( \varphi \) be the angle between the face \( ABD \) and the base plane \( ABC \) (see the picture). Choose the correct expression for \( \varphi \):
\( \cos\varphi=\frac13 \)
\( \mathrm{tg}\,\varphi=6\sqrt3 \)
\( \mathrm{tg}\,\varphi=3\sqrt2 \)
\( \cos\varphi=\frac23 \)

1103101207

Level: 
C
Let \( ABCD \) be a regular tetrahedron with an edge length of \( 5\,\mathrm{cm} \). Let \( \varphi \) be the angle between the edge \( AD \) and the base plane \( ABC \) (see the picture). Choose the correct expression for \( \varphi \):
\( \cos\varphi=\frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi=\sqrt3 \)
\( \mathrm{tg}\,\varphi=\frac{\sqrt3}3 \)
\( \cos\varphi=\sqrt3 \)

1103101206

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 10\,\mathrm{cm} \). Find the distance between the line \( AC \) and the line \( FD \) (see the picture).
\( 4\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)
\( 2\sqrt2\,\mathrm{cm} \)
\( 4\sqrt3\,\mathrm{cm} \)

1103101205

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with an edge \( AB \), where \( |AB| = 4\,\mathrm{cm} \), and an edge \( AV \), where \( |AV| = 8\,\mathrm{cm} \). Find the angle between the line \( AV \) and the base plane \( ABC \) (see the picture).
\( 60^{\circ} \)
\( 45^{\circ} \)
\( 72^{\circ} \)
\( 30^{\circ} \)

1103101204

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 6\,\mathrm{cm} \). Let \( \varphi \) be the angle between the plane \( AFV \) and the base plane \( ABC \) (see the picture). Choose the correct expression for \( \varphi \):
\( \mathrm{tg}\,\varphi=\sqrt3 \)
\( \sin\varphi=\sqrt3 \)
\( \mathrm{tg}\,\varphi=\frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi=\frac32 \)