Metric properties

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 \( \varphi \):
\( \mathrm{tg}\,\varphi=\sqrt3 \)
\( \sin\varphi=\sqrt3 \)
\( \mathrm{tg}\,\varphi=\frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi=\frac32 \)

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 \( S \).
\( 3\sqrt6\,\mathrm{cm} \)
\( 6\sqrt5\,\mathrm{cm} \)
\( 3\sqrt5\,\mathrm{cm} \)
\( 6\sqrt3\,\mathrm{cm} \)