Lines and planes: distances and angles

2010015607

Level: 
A
The rectangular box \( ABCDA'B'C'D' \) has edges of lengths \( |AB|=5\,\mathrm{cm} \) and \( |BC|=6\,\mathrm{cm} \). The distance between the center of the top face \(A'B'C'D'\) and the center of the bottom face \(ABCD\) is \(12\,\mathrm{cm}\). Find the length of the diagonal \(DC'\).
\( 13\,\mathrm{cm} \)
\( 6\sqrt5 \,\mathrm{cm} \)
\( \sqrt{119}\,\mathrm{cm} \)
\(6 \sqrt{3}\,\mathrm{cm} \)

2010015606

Level: 
A
The rectangular box \( ABCDA'B'C'D' \) has edges of lengths \( |AB|=4\sqrt3\,\mathrm{cm} \) and \( |BC|=8\,\mathrm{cm} \). The point \(S\) is the center of the lateral face \(ADD'A'\) (see the picture) and the length of the line segment \(B'S\) is \(10\,\mathrm{cm}\). Find the distance between the points \(A\) and \(A'\).
\( 12\,\mathrm{cm} \)
\( 6\,\mathrm{cm} \)
\( \sqrt{164}\,\mathrm{cm} \)
\( \sqrt{272}\,\mathrm{cm} \)

2010015605

Level: 
A
The rectangular box \( ABCDA'B'C'D' \) has edges of lengths \( |AB|=6\,\mathrm{cm} \) and \( |BC|=8\,\mathrm{cm} \). The point \(S\) is the center of the base \(ABCD\) (see the picture) and the length of the line segment \(A'S\) is \(13\,\mathrm{cm}\). Find the distance between the points \(A\) and \(A'\).
\( 12\,\mathrm{cm} \)
\( \sqrt{194}\,\mathrm{cm} \)
\( \sqrt{69}\,\mathrm{cm} \)
\( 4\sqrt{10}\,\mathrm{cm} \)

2010015604

Level: 
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 4\,\mathrm{cm} \). The height of the pyramid is \( 6\,\mathrm{cm} \). Find the distance between the point \( A \) and the point \( S_{VB} \), where \( S_{VB} \) is the midpoint of the edge \( VB \).
\( \sqrt{19}\,\mathrm{cm} \)
\( \sqrt{35}\,\mathrm{cm} \)
\( 3\sqrt{3}\,\mathrm{cm} \)
\( \sqrt{5}\,\mathrm{cm} \)

2010015603

Level: 
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 12\,\mathrm{cm} \). The lateral edge of the pyramid is \( 10\,\mathrm{cm} \). Find the distance between the point \( V \) and the base \( ABCD \).
\( 8\,\mathrm{cm} \)
\( \sqrt{34}\,\mathrm{cm} \)
\( \sqrt{44}\,\mathrm{cm} \)
\( \sqrt{11}\,\mathrm{cm} \)

2010015602

Level: 
B
The base of a pyramid is a square with the side of \(4\, \mathrm{cm}\). The height of the pyramid is \(8\, \mathrm{cm}\). Find the angle between the lateral edge of the pyramid and the base plane. Round your result to two decimal places.
\( 70.53^{\circ} \)
\( 19.47^{\circ} \)
\( 75.96^{\circ} \)

2010015601

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 lines \( AD' \) and \( CD' \). Round the result to two decimal places.
\( 31.31^{\circ} \)
\( 58.69^{\circ} \)
\( 16.70^{\circ} \)
\( 20.57^{\circ} \)

Square Pyramid -- Angles

Submitted by ladislav.foltyn on Sat, 06/01/2019 - 12:23
Question: 
\vspace{-2em} \begin{minipage}{0.55\linewidth} The base $ABCD$ of a square pyramid $ABCDV$ has an edge of length $a$, and the lateral face is an equilateral triangle (see the picture). Let $S$ be the midpoint of the base $ABCD$ and let $P$ be the midpoint of the edge $AV$. Find the angle between \end{minipage} \hfill \begin{minipage}{0.4\linewidth} \obrMsr[x=3cm,y=3cm,z=0.3cm]{-1}2{-1}2 { \footnotesize \pgfmathsetmacro{\cubex}{1} \pgfmathsetmacro{\cubey}{1} \pgfmathsetmacro{\cubez}{2} \coordinate (A) at (0,0,0); \coordinate (B) at (\cubex,0,0); \coordinate (C) at (\cubex.2,0,\cubez); \coordinate (D) at (0.2,0,\cubez); \coordinate (V) at (0.6,0.7,1); \coordinate (P) at ($(A)!0.5!(V)$); \draw[thick,dashed] (A) -- (D) node [yshift=4pt,xshift=-6pt]{$D$} -- (C) node [yshift=-5pt,xshift=5pt]{$C$}; \draw[dashed] (A) -- (C); \draw[dashed] (B) -- (D); \draw (0.6,0,1) node [below,xshift=-2pt,yshift=1pt]{$S$}; \draw[thick] (A) node [yshift=-5pt,xshift=-5pt]{$A$} -- (B) node [yshift=-6pt,xshift=3pt]{$B$} --(C); \draw[thick] (A) -- (V) node [above]{$V$}; \draw[thick] (B) -- (V); \draw[thick] (C) -- (V); \draw[thick,dashed] (D) -- (V); \draw[dashed] (0.6,0,1) -- (V); \begin{scope}[thick] \obrKrizek[2pt]{P}{above left}{P} \end{scope} } \end{minipage}