Lines and planes: distances and angles

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 \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2\sqrt{2}} {6} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 50^{\circ }29^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {6}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 36^{\circ }52^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {2\sqrt{10}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 35^{\circ }6^{\prime}\)

9000153603

Level: 
B
Give a verbal description to the angle shown in the picture.
The angle between two opposite triangular faces.
The angle between a triangular face and the base.
The angle between two edges in the same triangular face.
The angle between two triangular faces having a common edge.

9000153605

Level: 
B
Give a verbal description to the angle shown in the picture.
The angle between opposite edges.
The angle between a triangular face and an edge from the opposite triangular face.
The angle between two opposite triangular faces.
The angle between two triangular faces having a common edge.

9000153606

Level: 
B
Give a verbal description to the angle shown in the picture.
The angle between the edge on triangular face and the base edge from the same face.
The angle between the triangular face and a base edge not in this face.
The angle between two triangular faces having a common edge.
The angle between a triangular face and square base.

9000153701

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 \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 71^{\circ }34^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {6}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 36^{\circ }52^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {2\sqrt{10}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 35^{\circ }6^{\prime}\)

9000128808

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). Find the angle between the planes \(ADV \) and \(BCV \). Round to two decimal places.
\(73.74^{\circ }\)
\(36.87^{\circ }\)
\(61.93^{\circ }\)

9000128802

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). The point \(M\) is the middle of the side \(CV \). Find the distance between the point \(M\) and the line \(BC\).
\(\frac{5} {2}\, \mathrm{cm}\)
\(\frac{\sqrt{34}} {2} \, \mathrm{cm}\)
\(\frac{\sqrt{7}} {2} \, \mathrm{cm}\)