Metric properties

1103055806

Level: 
A
The rectangular box \( ABCDEFGH \) shown in the picture has edges of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \) and \( |AE|=8\,\mathrm{cm} \). Find the distance between the lines \(ES_{FG} \) and \( DS_{BC} \), where \( S_{FG} \) is the center of \(FG\) and \( S_{BC} \) is the center of \(BC\).
\( 8 \,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
\( 4\sqrt5\,\mathrm{cm} \)
\( 8\sqrt2\,\mathrm{cm} \)

1103055805

Level: 
A
The rectangular box \( ABCDEFGH \) shown in the picture has edges of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \) and \( |AE|=8\,\mathrm{cm} \). Find the distance between the lines \( AB \) and \( HG \).
\( 4\sqrt5\,\mathrm{cm} \)
\( 8\sqrt2\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

1103055804

Level: 
A
The rectangular box \( ABCDEFGH \) shown in the picture has edges of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \) and \( |AE|=8\,\mathrm{cm} \). Find the distance between the lines \( AH \) and \( FC \).
\( 6\,\mathrm{cm} \)
\( 2\sqrt{13}\,\mathrm{cm} \)
\( 0 \,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)

1103055803

Level: 
A
The rectangular box \( ABCDEFGH \) has edges of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=3\,\mathrm{cm} \) and the face diagonal of the length \( |BG|=5\,\mathrm{cm} \). Find the distance between the center of the top face \( EFGH \) and the center of the bottom base \( ABCD \).
\( 4\,\mathrm{cm} \)
\( 6\,\mathrm{cm} \)
\( 7.5\,\mathrm{cm} \)
\( \sqrt{61}\,\mathrm{cm} \)

1103055802

Level: 
A
The rectangular box \( ABCDEFGH \) has edges of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \) and \( |AE|=8\,\mathrm{cm} \). The point \( S \) is the center of the left side face \( ADHE \) as seen in the picture. Find the distance between the point \( F \) and the point \( S \).
\( 2\sqrt{14}\,\mathrm{cm} \)
\( 2\sqrt{29}\,\mathrm{cm} \)
\( 2\sqrt{17}\,\mathrm{cm} \)
\( 4\sqrt{2}\,\mathrm{cm} \)

1103055801

Level: 
A
The rectangular box \( ABCDEFGH \) has edges of lengths \( |AB|=6\,\mathrm{cm} \), \( |BC|=4\,\mathrm{cm} \) and \( |AE|=8\,\mathrm{cm} \). The point \( S \) is the center of the base \( ABCD \) as seen in the picture. Find the distance between the point \( E \) and the point \( S \).
\( \sqrt{77}\,\mathrm{cm} \)
\( 3\sqrt{10}\,\mathrm{cm} \)
\( 2\sqrt{26}\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)

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 \( BDD' \).
\( 3 \)
\( \sqrt3 \)
\( 2\sqrt3 \)
\( \frac{\sqrt3}2 \)

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 \( CD' \).
\( 3\sqrt3 \)
\( 6 \)
\( 6\sqrt3 \)
\( \frac32\sqrt3 \)

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.
\( 57^{\circ} \)
\( 53.13^{\circ} \)
\( 33^{\circ} \)
\( 72.01^{\circ} \)