Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex and \( K \), \( L \), and \( M \) are the midpoints of its edges \( AD \), \( BC \), and \( CV \) respectively. What is the mutual position of planes \( BVK \) and \( DLM \)?
Let \( ABCDEFGH \) be a cube with \( K \), and \( L \) being the midpoints of its edges \( AE \) and \( CG \), and let \( M \) be the centre of its face \( ABFE \). What is the mutual position of planes \( BCE \), \( ADF \), and \( KLM \)?
three mutually intersecting planes which share one common line
three mutually intersecting planes which share one common point
two planes are parallel with the third intersecting them in distinct parallel lines
Let \( ABCDEFGH \) be a cube and \( X \) be the midpoint of its edge \( AE \). What is the cross-section of the cube if we slice it with a plane \( BGX \)?
a quadrilateral \( BGPX \) with \( P \) being the midpoint of the edge \( EH \)
a quadrilateral \( BGHX \)
a triangle \( BGX \)
a quadrilateral \( BGPX \) with \( P \) being the midpoint of the edge \( DH \)
Let \( ABCDEFGH \) be a cube and let \( X \), \( Y \), and \( Z \) be the midpoints of edges \( AB \), \( AE \), and \( CG \) respectively. What is the cross-section of the cube if we slice it with a plane \( XYZ \)?
a hexagon \( XLZKMY \) with points \( L \), \( K \), and \( M \) lying on edges \( BC \), \( GH \), and \( EH \) respectively
a pentagon \( XLZKY \) with points \( L \) and \( K \) lying on edges \( BC \) and \( GH \) respectively
a triangle \( XYZ \)
a quadrilateral \( XZKY \) with \( K \) being the midpoint of the edge \( GH \)
Let \( ABCDEFGH \) be a cube with \( K \) and \( L \) being the midpoints of edges \( AE \) and \( AB \) respectively, and let \( M \) be the midpoint of the face diagonal \( EG \). What is the cross-section of the cube if we slice it with a plane \( KLM \)?
a pentagon \( KLPQR \) with points \( P \), \( Q \), and \( R \) lying on edges \( BC \), \( FG \), and \( EH \) respectively
a triangle \( KLM \)
a pentagon \( KLPQM \) with points \( P \) and \( Q \) lying on edges \( BC \) and \( FG \) respectively
a quadrilateral \( KLMR \) with point \( R \) lying on the edge \( EH \)
Let \( ABCDV \) be a rectangle based-pyramid and \( V \) its apex. The pyramid is sliced by a plane \( EFG \) which is defined by:
\begin{align*}
E&\in BC\ \wedge\ |BE|=2|CE|, \\
F&\in AV\ \wedge\ |AF|=2|VF|, \\
G&\in DV\ \wedge\ |DG|=2|VG|
\end{align*}
(see the picture). What is the cross-section of the pyramid if we slice it with the plane \( EFG \)?
Let \( ABCDV \) be a rectangle based-pyramid and \( V \) its apex. The pyramid is sliced by a plane \( XYZ \) which is defined by:
\begin{align*}
X&\text{ is the midpoint of the edge }AD,\\
Y&\in CD\ \wedge\ |DY|=3|CY|,\\
Z&\in BV\ \wedge\ |BZ|=3|VZ|
\end{align*}
(see the picture). What is the cross-section of the pyramid if we slice it with the plane \( XYZ \)?
a pentagon \( XYKZL \) with points \( K \) and \( L \) lying on the edges \( CV \) and \( AV \)
a triangle \( XYZ \)
a quadrilateral \( XYZL \) with point \( L \) lying on the edge \( AV \)
a quadrilateral \( XYKZ \) with point \( K \) lying on the edge \( CV \)
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where:
\begin{align*}
X&\text{ lays on a ray }BC\text{ and }|BX|=1.5|BC|,\\
Y&\text{ lays on a ray }HE\text{ and }|HY|=1.5|HE|
\end{align*}
(see the picture). The points of intersection of the line \( XY \) with the surface of the cube lay: