Positional properties

1103059505

Level: 
C
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 \( DH \)

1103059506

Level: 
C
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 \( GH \)

1103059507

Level: 
C
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 \( EH \)

1103059601

Level: 
C
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 \( EFG \)?
a trapezium \( BCGF \)
a triangle \( EFG \)
a triangle \( AEV \)
a pentagon \( ABEGF \)

1103059602

Level: 
C
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 \( CV \)

1103059603

Level: 
C
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:
on the sides \( ABFE \) and \( DCGH \)
on the side \( ABFE \) and the edge \( CG \)
on the edges \( AE \) and \( CG \)
on the sides \( ADHE \) and \( BCGF \)

1103059604

Level: 
C
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }DH\text{ and }|DX|=1.5|DH|,\\ Y&\text{ lays on a ray }DB\text{ and }|DB|=|BY| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the cube lay:
on the side \( EFGH \) and the edge \( BF \)
on the edges \( EF \) and \( BF \)
on the sides \( EFGH \) and \( ABCD \)
on the edges \( HG \) and \( BF \)

1103059605

Level: 
C
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }CB\text{ and }|CX|=1.5|BC|,\\ Y&\text{ lays on a ray }EH\text{ and }|EY|=1.5|EH| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the cube lay:
on the sides \( ABFE \) and \( DCGH \)
on the sides \( EFGH \) and \( ABCD \)
on the side \( ABCD \) and the edge \( HG \)
on the edges \( HG \) and \( AB \)

1103059606

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex, and let \( XY \) be a line where: \begin{align*} X&\text{ lays on the side }AV\text{ and }|AX|=|XV|,\\ Y&\text{ lays on a ray }DC\text{ and }|DY|=1.5|DC| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the pyramid are:
the point \( X \) and a point on the pyramid face \( BCV \)
the point \( X \) and a point on the pyramid face \( DCV \)
the point \( X \) and a point on the pyramid edge \( CV \)
the point \( X \) only

1103059607

Level: 
C
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex, and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }BA\text{ and }|BA|=|AX|,\\ Y&\text{ lays on the height }SV\text{ and }|SY|=|YV|,\\ S&\text{ is the centre of the pyramid base} \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the pyramid lay:
on the pyramid faces \( ADV \) and \( BCV \)
on the pyramid faces \( DCV \) and \( ABV \)
on the pyramid face \( ADV \) and its edge \( CV \)
on the pyramid edges \( AV \) and \( CV \)