Metric properties

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.
\( 2\,\mathrm{cm} \)
\( 2.5\,\mathrm{cm} \)
\( \frac{\sqrt{52}}2\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)

1103025302

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 \( S_{VB}S_{VC}\) and the line \( BC \). The points $S_{VB}$ and $S_{VC}$ are the midpoints of $VB$ and $VC$, respectively.
\( 2.5\,\mathrm{cm} \)
\( 2\,\mathrm{cm} \)
\( \frac{\sqrt{52}}2\,\mathrm{cm} \)
\( \frac{25}2\,\mathrm{cm} \)

1103025301

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 point \( V \) and the line \( BC \).
\( 5\,\mathrm{cm} \)
\( \sqrt{52}\,\mathrm{cm} \)
\( 25\,\mathrm{cm} \)
\( \sqrt{10}\,\mathrm{cm} \)

1103018804

Level: 
A
Choose the correct verbal description of the angle shown in the picture, where the point \(S_{EF}\) is the midpoint of the edge \(EF\).
The angle between the line \(AS_{EF}\) and the plane \(BCG\) (right side face).
The angle between the line \(AS_{EF}\) and the plane \(EFG\) (top face).
The angle between the line \(AS_{EF}\) and the plane \(DCG\) (back face).
The angle between the line \(AS_{EF}\) and the plane \(ABF\) (front face).

1103018802

Level: 
A
Choose the correct verbal description of the angle shown in the picture.
The angle between a space diagonal of a cube and its face diagonal.
The angle between a space diagonal of a cube and its edge.
The angle between two face diagonals of a cube.
The angle between a face diagonal of a cube and its edge.

1103018801

Level: 
A
Choose the correct verbal description of the angle shown in the picture:
The angle between two space diagonals of a cube.
The angle between a space diagonal of a cube and its edge.
The angle between two face diagonals of a cube.
The angle between a space diagonal of a cube and a face diagonal.

9000153706

Level: 
B
The picture shows a square pyramid. The side of a base square is \(a = 4\; \mathrm{cm}\) and the height of the pyramid is \(v = 6\; \mathrm{cm}\). Find the angle \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{2\sqrt{10}} {2} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 72^{\circ }27^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 71^{\circ }34^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2\sqrt{2}} {6} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 50^{\circ }29^{\prime}\)

9000153705

Level: 
B
The picture shows a square pyramid. The side of a base square is \(a = 4\; \mathrm{cm}\) and the height of the pyramid is \(v = 6\; \mathrm{cm}\). Find the angle \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2\sqrt{2}} {6} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 50^{\circ }29^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {6}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 36^{\circ }52^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {2\sqrt{10}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 35^{\circ }6^{\prime}\)

9000153702

Level: 
B
The picture shows a square pyramid. The side of a base square is \(a = 4\; \mathrm{cm}\) and the height of the pyramid is \(v = 6\; \mathrm{cm}\). Find the angle \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 71^{\circ }34^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {6}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 36^{\circ }52^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{2\sqrt{10}} {2} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 72^{\circ }27^{\prime}\)