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5 problems match “law of cosines”. Clear

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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 with circumcenter vector ●●●○○ 50 ptsIn triangle $ABC$, $AB = 3$ and $AC = 5$. Let $O$ be the circumcenter of triangle $ABC$. Find $\overrightarrow{OA} \cdot \overrightarrow{BC}$.
3. Cosine squared in a heptagon ●●●●○ 100 ptsLet $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of triangle $ABD$. Find $\cos^2 \angle GOM$. Give your answer as a decimal.
4. Side length of a square tilted to a plane ●●●●○ 100 ptsGiven a square $ABCD$ with side length $a$, vertex $A$ lies in plane $\beta$, and the other vertices are on the same side of plane $\beta$. The distances from points $B$ and $D$ to plane $\beta$ are 1 and 2, respectively. If the angle between plane $ABCD$ and plane $\beta$ is $30^\circ$, find $a$. Round to two decimal places.
5. Shortest network connecting regular pentagon vertices ●●●●○ 100 ptsA regular pentagon has side length 1 cm. Find the total length, in cm, of the shortest network of line segments that connects all five vertices of the pentagon. Give your answer in centimetres, rounded to two decimal places.