Lines and planes: distances and angles

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

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).
\( 4\sqrt3\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( 8\sqrt3\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)

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).
\( 2\sqrt{19}\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( \left(8+2\sqrt{3}\right)\,\mathrm{cm} \)
\( 4\sqrt{5}\,\mathrm{cm} \)

1103056006

Level: 
A
The cube \( ABCDEFGH \) shown in the picture has edges of length \( a=6\,\mathrm{cm} \). Find the distance between the point \( B \) and the line \( EG \).
\( 3\sqrt6\,\mathrm{cm} \)
\( 6\sqrt3\,\mathrm{cm} \)
\( 3\sqrt5\,\mathrm{cm} \)
\( \frac{3\sqrt6}2\,\mathrm{cm} \)

1103056004

Level: 
A
The cube \( ABCDEFGH \) shown in the picture has edges of length \( a=6\,\mathrm{cm} \). Let \( S_1 \) be the midpoint of the diagonal \( ED \) and let \( S_2 \) be the midpoint of the diagonal \( CH \). Find the distance between the points \( S_1 \) and \( S_2 \).
\( 3\sqrt2\,\mathrm{cm} \)
\( 6\sqrt2\,\mathrm{cm} \)
\( \sqrt2\,\mathrm{cm} \)
\( 6\sqrt3\,\mathrm{cm} \)

1103056003

Level: 
A
The cube \( ABCDEFGH \) shown in the picture has edges of length \( a=6\,\mathrm{cm} \). Let \( S \) be the midpoint of the edge \( FG \). Find the distance between the points \( E \) and \( S \).
\( 3\sqrt5\,\mathrm{cm} \)
\( 6\sqrt5\,\mathrm{cm} \)
\( \sqrt5\,\mathrm{cm} \)
\( 6\sqrt3\,\mathrm{cm} \)

1103056002

Level: 
A
The cube \( ABCDEFGH \) shown in the picture has edges of length \( a=6\,\mathrm{cm} \). Let \( S \) be the midpoint of the base \( ABCD \). Find the distance between the points \( H \) and \( S \).
\( 3\sqrt6\,\mathrm{cm} \)
\( 6\sqrt5\,\mathrm{cm} \)
\( 3\sqrt5\,\mathrm{cm} \)
\( 6\sqrt3\,\mathrm{cm} \)