Search problems

4 problems match “regular tetrahedron”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. Angle CBD in a tetrahedron ●●○○○ 20 ptsIn tetrahedron $ABCD$, $BC=1$, $BD=2$, $\angle ABC=90^{\circ}$, and $\overrightarrow{BC}\cdot\overrightarrow{DA}=-\sqrt{3}$. Find $\angle CBD$ in degrees.
2. Dot product in a regular tetrahedron ●●●○○ 50 ptsGiven a regular tetrahedron $P-ABC$ with edge length $1$, points $E, F, G, H$ lie on edges $PA, PB, CA, CB$ respectively. If $\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}$ and $\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}$, find $\overrightarrow{EG} \cdot \overrightarrow{FH}$. Round to two decimal places.
3. Sum of squared tangents in a tetrahedron section ●●●●○ 100 ptsA regular tetrahedron is intersected by a plane that is perpendicular to one of its faces and meets that face along a line $e$. The intersection lines of this plane with the other three face planes of the tetrahedron form angles $\varphi_1, \varphi_2, \varphi_3$ with $e$. Determine the value of $\tan^2 \varphi_1+\tan^2 \varphi_2+\tan^2 \varphi_3$.
4. Angle between faces of an equifacial tetrahedron ●●●●○ 100 ptsLet $ABCD$ be a tetrahedron in which opposite edges are equal. The angle between planes $ABD$ and $BCD$ is $90^\circ$, and the angle between planes $BCD$ and $CAD$ is $60^\circ$. Find the angle between planes $CAD$ and $ABD$ in degrees.