9000121006
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(BG\)
and \(GD\).
\(60^{\circ }\)
\(45^{\circ }\)
\(36.87^{\circ }\)
\(53.13^{\circ }\)
Metric Properties
9000121007
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(S_{BE}S_{AH}\)
and \(HC\), where
\(S_{BE}\) and
\(S_{AH}\) are centers of
the segments \(BE\)
and \(AH\),
respectively.
\(60^{\circ }\)
\(26.57^{\circ }\)
\(45^{\circ }\)
\(53.13^{\circ }\)
9000121009
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(S_{HD}S_{FC}\)
and \(AB\), where
the points \(S_{HD}\)
and \(S_{FC}\) are
centers of \(HD\)
and \(FC\),
respectively.
\(26.57^{\circ }\)
\(45^{\circ }\)
\(54.74^{\circ }\)
\(60^{\circ }\)
9000121010
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(ES_{CG}\)
and \(AG\), where
\(S_{CG}\) is the center
of the \(CG\).
\(54.74^{\circ }\)
\(19.47^{\circ }\)
\(35.26^{\circ }\)
\(60^{\circ }\)
9000120304
Level:
C
The side of a regular hexagonal prism
\(ABCDEFA'B'C'D'E'F'\) is
\(a = 3\, \mathrm{cm}\) and the height
\(v = 8\, \mathrm{cm}\). Find the length
of the diagonal \(AD'\).
\(10\, \mathrm{cm}\)
\(\sqrt{73}\, \mathrm{cm}\)
\(\sqrt{82}\, \mathrm{cm}\)
\(2\sqrt{8}\, \mathrm{cm}\)
\(2\sqrt{6}\, \mathrm{cm}\)
9000120303
Level:
A
Identify a valid relation involving the angle
\(\alpha \)
defined as an angle between a solid diagonal and a face diagonal through the same
vertex in a cube.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{\sqrt{2}}
{2} \)
\(\sin \alpha = \frac{\sqrt{3}}
{2} \)
\(\cos \alpha = \frac{\sqrt{5}}
{3} \)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \sqrt{3}\)
\(\alpha = 45^{\circ }\)
9000121003
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(BS_{DH}\)
and \(AE\), where
the point \(S_{DH}\) is the
center of the edge \(DH\).
\(70.53^{\circ }\)
\(29.47^{\circ }\)
\(35.26^{\circ }\)
\(54.74^{\circ }\)
9000121004
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(S_{AE}S_{HC}\)
and \(S_{HC}S_{BF}\),
where \(S_{AE}\),
\(S_{HC}\) and
\(S_{BF}\) are the centers
of the segments \(AE\),
\(HC\) and
\(BF\),
respectively.
\(53.13^{\circ }\)
\(26.57^{\circ }\)
\(60^{\circ }\)
\(36.87^{\circ }\)
9000121005
Level:
A
In the cube \(ABCDEFGH\) find the
angle between the lines \(AS_{BC}\)
and \(AD\), where
\(S_{BC}\) is the center
of the edge \(BC\).
\(63.43^{\circ }\)
\(26.57^{\circ }\)
\(53.13^{\circ }\)
\(36.87^{\circ }\)
9000120302
Level:
A
A cuboid has sides \(a = 5\, \mathrm{cm}\),
\(b = 8\, \mathrm{cm}\), and
\(c = \sqrt{111}\, \mathrm{cm}\). Find the length of the cuboid’s space diagonal \(u\) (see the picture).
\(10\sqrt{2}\, \mathrm{cm}\)
\(\sqrt{222}\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(2\sqrt{10}\, \mathrm{cm}\)
\(5\sqrt{7}\, \mathrm{cm}\)