2000006503
Level:
A
A plane and a point are given. The point is not lying in the plane. How many lines perpendicular to this plane pass through the given point?
\(1\)
\(2\)
\(0\)
infinitely many
Positional properties
2000006502
Level:
A
How many pairs of intersecting planes are created by sides of a cuboid?
\(12\)
\(9\)
\(6\)
\(3\)
2000006501
Level:
A
How many pairs of intersecting planes are created by sides of a cube?
\(12\)
\(6\)
\(9\)
\(3\)
Relative Position of Lines
Submitted by ladislav.foltyn on Mon, 07/01/2019 - 08:48Relative Position of Circles
Submitted by ladislav.foltyn on Sun, 05/05/2019 - 16:52Question:
\kern -2em
Let $p$ be a line containing the points $S_4$, $S_1$, $S_3$, $S_2$ in this order, where $|S_4 S_1|= 1\,\mathrm{cm}$, $|S_1 S_3|= 1.5\,\mathrm{cm}$, and $|S_3 S_2|= 3.5\,\mathrm{cm}$.
\smallskip
Further, let $k_1$, $k_2$, $k_3$, $k_4$, and $k_5$ be circles with the centers $S_1$, $S_2$, $S_3$, $S_4$, and $S_2$ (again) and radii $r_1=3\,\mathrm{cm}$, $r_2=2\,\mathrm{cm}$, $r_3=1.5\,\mathrm{cm}$, $r_4=1.5\,\mathrm{cm}$, and $r_5=8\,\mathrm{cm}$ respectively. Determine the relative position of the circles.
\kern 6em
Points, Segments, Rays and Lines
Submitted by ladislav.foltyn on Fri, 04/19/2019 - 18:26Question:
Find the correct symbolic notation of the statements.
Pairs of Lines and Pairs of Planes
Submitted by ladislav.foltyn on Wed, 03/06/2019 - 18:46Question:
The picture shows a cube $ABCDEFGH$ with points $K$ and $L$ being the centers of edges $AE$ and $BF$. Determine the mutual positions of lines and planes.
1103059607
Level:
B
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 \( CV \)
1103059606
Level:
B
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
1103059605
Level:
B
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 \( AB \)