Analytical space geometry

1103212902

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the face \( ABFE \), and let \( K \) and \( L \) be the midpoints of edges \( DH \) and \( CG \) consecutively. Find the standard equation of a plane \( \alpha \) passing through the points \( A \), \( B \) and \( L \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)

1103212903

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Find an angle \( \varphi \) between the plane \( \alpha \) passing through the points \( E \), \( D \) and \( C \) and the straight line \( AF \). Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.
\( \varphi = 30^{\circ} \)
\( \varphi = 15^{\circ} \)
\( \varphi = 45^{\circ} \)
\( \varphi = 60^{\circ} \)

1103212904

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with a bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the edge \( AD \). Find the standard equation of the plane \( \alpha \) passing through the points \( B \), \( V \) and \( C \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)

1103212905

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with its bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Find the parametric equations of an intersection line \( p \) of planes \( \alpha \) and \( \beta \), where \( \alpha \) passes through the points \( B \), \( C \) and \( V \), and \( \beta \) passes through the points \( A \), \( D \) and \( V \). What is the measure of an angle \( \varphi \) between the planes \( \alpha \) and \( \beta \). Round \( \varphi \) to the nearest minute.
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=0;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3+t, &\\ z&=6+2t;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)

1103233601

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find its perpendicular height. \[ \] Hint: Find e.g. the distance between the point $F$ and the plane $ACH$.
$\frac{2\sqrt3}3$
$\frac{\sqrt3}3$
$\frac{2\sqrt6}3$
$\frac23$

1103233602

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the distance between the opposite edges of this tetrahedron.\[ \] Hint: A tetrahedron’s opposite edges lie on skew lines. Their distance is the same as the distance of the midpoint of one edge from the opposite edge.
$1$
$\sqrt3$
$\frac{\sqrt3}2$
$\frac{\sqrt5}2$

1103233603

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the angle between its faces and round the number to the nearest minute.
$70^{\circ}32'$
$54^{\circ}44'$
$45^{\circ}$
$51^{\circ}4'$

2010008703

Level: 
C
A straight line \( q \) is given by the points \( K=[6;6;7] \) and \( L=[4;0;2] \) (see the picture). Find the parametric equations of the line \( q' \) that is symmetrical to the line \( q \) in the plane symmetry across the coordinate \( xz \)-plane.
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=-6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=-6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)