1103018906
Level:
A
In the cube \( ABCDEFGH \), find the angle between two face diagonals \( EG \) and \( EB \).
\( 60^{\circ} \)
\( 90^{\circ} \)
\( 45^{\circ} \)
\( 30^{\circ} \)
Lines and planes: distances and angles
1103018905
Level:
A
In the cube \( ABCDEFGH \) with \( S_{AC} \) being the midpoint of the diagonal \( AC \), let \( \varphi \) be the angle between the line \( EG \) and the line \( GS_{AC} \). Choose the correct expression for \( \varphi \):
\( \mathrm{tg}\,\varphi = \sqrt2 \)
\( \mathrm{sin}\,\varphi = \frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{cos}\,\varphi = \frac{\sqrt6}3 \)
1103018904
Level:
A
In the cube \( ABCDEFGH \), let \( S_{FG} \) be the midpoint of the edge \( FG \). Find the angle between the lines \( BS_{FG} \) and \( BF \). Round the result to two decimal places.
\( 26.57^{\circ} \)
\( 22.5^{\circ} \)
\( 45^{\circ} \)
\( 54.74^{\circ} \)
1103018903
Level:
A
In the cube \( ABCDEFGH \) with \( S_{AC} \) being the midpoint of the diagonal \( AC \), let \( \varphi \) be the angle between the line \( ES_{AC} \) and the bottom face \( ABCD \). Choose the correct expression for \( \varphi \).
\( \mathrm{tg}\,\varphi = \sqrt2 \)
\( \mathrm{sin}\,\varphi = \frac{\sqrt2}3 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{cos}\,\varphi = \frac{\sqrt6}3 \)
1103018902
Level:
A
Let \( \varphi \) bet the angle between a space diagonal of a cube and its face diagonal. Choose the correct expression for \( \varphi \).
\( \mathrm{tg}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{sin}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt6}3 \)
1103018901
Level:
A
In the cube \( ABCDEFGH \), find the angle between the space diagonals \( AG \) and \( BH \). Round the result to two decimal places.
\( 70.53^{\circ} \)
\( 45^{\circ} \)
\( 54.74^{\circ} \)
\( 35.27^{\circ} \)
1103025306
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 6\,\mathrm{cm} \). The height of the pyramid is \( 3\sqrt2\,\mathrm{cm} \). Find the distance between the point \( A \) and the line \( BV \) (see the picture).
\( 3\sqrt3\,\mathrm{cm} \)
\( 3\sqrt2\,\mathrm{cm} \)
\( \frac32\sqrt3\,\mathrm{cm} \)
\( 3\sqrt6\,\mathrm{cm} \)
1103025305
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 4\,\mathrm{cm} \). The height of the pyramid is \( 6\,\mathrm{cm} \). Find the distance between the point \( A \) and the point \( S_{VC} \), where \( S_{VC} \) is the midpoint of the edge \( VC \).
\( 3\sqrt{3}\,\mathrm{cm} \)
\( 4\sqrt{3}\,\mathrm{cm} \)
\( \sqrt{10}\,\mathrm{cm} \)
\( 3\sqrt{10}\,\mathrm{cm} \)
1103025304
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 8\,\mathrm{cm} \). The height of the pyramid is \( 9\,\mathrm{cm} \). Find the distance between the line \( S_{VA}S_{VD} \) and the line \( BC \). The points $S_{VA}$ and $S_{VD}$ are the midpoints of $VA$ and $VD$, respectively.
\( 7.5\,\mathrm{cm} \)
\( \frac23\sqrt{97}\,\mathrm{cm} \)
\( \frac{\sqrt{97}}2\,\mathrm{cm} \)
\( \sqrt{17}\,\mathrm{cm} \)
1103025303
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 6\,\mathrm{cm} \). The height of the pyramid is \( 4\,\mathrm{cm} \). Find the distance between the line \( S_{VA}S_{VC} \) and the line \( AC \). The points $S_{VA}$ and $S_{VC}$ are the midpoints of $VA$ and $VC$, respectively.
\( 2\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)
\( \frac{\sqrt{52}}2\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)