Metric properties

1103101201

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 8\,\mathrm{cm} \). Find the distance between the point \( V \) and the line \( BC \) (see the picture).
\( 2\sqrt{19}\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( \left(8+2\sqrt{3}\right)\,\mathrm{cm} \)
\( 4\sqrt{5}\,\mathrm{cm} \)

1103101202

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 8\,\mathrm{cm} \). Find the distance between the line \( AB \) and the line \( ED \) (see the picture).
\( 4\sqrt3\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( 8\sqrt3\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)

1103101203

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 4\sqrt3\,\mathrm{cm} \). Find the angle between the lines \( FV \) and \( CV \) (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 \( \varphi \):
\( \mathrm{tg}\,\varphi=\sqrt3 \)
\( \sin\varphi=\sqrt3 \)
\( \mathrm{tg}\,\varphi=\frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi=\frac32 \)

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).
\( 60^{\circ} \)
\( 45^{\circ} \)
\( 72^{\circ} \)
\( 30^{\circ} \)

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).
\( 4\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)
\( 2\sqrt2\,\mathrm{cm} \)
\( 4\sqrt3\,\mathrm{cm} \)

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 \( \varphi \):
\( \cos\varphi=\frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi=\sqrt3 \)
\( \mathrm{tg}\,\varphi=\frac{\sqrt3}3 \)
\( \cos\varphi=\sqrt3 \)

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 \( \varphi \):
\( \cos\varphi=\frac13 \)
\( \mathrm{tg}\,\varphi=6\sqrt3 \)
\( \mathrm{tg}\,\varphi=3\sqrt2 \)
\( \cos\varphi=\frac23 \)

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).
\( \sqrt6\,\mathrm{cm} \)
\( \frac{3\sqrt3}2\,\mathrm{cm} \)
\( \frac{\sqrt3}2\,\mathrm{cm} \)
\( 3\sqrt2\,\mathrm{cm} \)

1103107001

Level: 
C
Let \( ABCDEFA'B'C'D'E'F' \) be a regular hexagonal prism with the base edge length of \( 4\,\mathrm{cm} \) and the height of \( 8\,\mathrm{cm} \). Find the angle between the lines \( FC' \) and \( CF' \) (see the picture).
\( 90^{\circ} \)
\( 45^{\circ} \)
\( 60^{\circ} \)
\( 72^{\circ} \)