Lines and planes: distances and angles

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

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 \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 71^{\circ }34^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2} {6}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 36^{\circ }52^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{2\sqrt{10}} {2} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 72^{\circ }27^{\prime}\)

9000153703

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

9000153704

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

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

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 \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{2\sqrt{10}} {2} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 72^{\circ }27^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2\sqrt{2}}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 64^{\circ }46^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{6} {2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 71^{\circ }34^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi } {2} = \frac{2\sqrt{2}} {6} \mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 50^{\circ }29^{\prime}\)

1103025401

Level: 
C
A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the angle between the lines \( AD' \) and \( BD' \). Round the result to two decimal places.
\( 17.46^{\circ} \)
\( 72.54^{\circ} \)
\( 16.70^{\circ} \)
\( 20.57^{\circ} \)

1103025402

Level: 
C
A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the angle between the plane \( ABD' \) and the base plane \( ABC \). Round the result to two decimal places.
\( 57^{\circ} \)
\( 53.13^{\circ} \)
\( 33^{\circ} \)
\( 72.01^{\circ} \)

1103025403

Level: 
C
A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the distance between the lines \( FA' \) and \( CD' \).
\( 3\sqrt3 \)
\( 6 \)
\( 6\sqrt3 \)
\( \frac32\sqrt3 \)

1103025404

Level: 
C
A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the distance between the planes \( AEE' \) and \( BDD' \).
\( 3 \)
\( \sqrt3 \)
\( 2\sqrt3 \)
\( \frac{\sqrt3}2 \)