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4 problems match “distance in 3D”. Clear

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1. Medians of a triangle in space ●●○○○ 20 ptsTriangle $ABC$ has vertices $A(0, 0, 6)$, $B(0, 4, 0)$ and $C(6, 0, 0)$. Find the lengths of the three medians of the triangle. Give each length rounded to two decimal places, listed in increasing order.
2. Area of triangle BDE in space ●●●○○ 50 ptsPoints $A$, $B$, $C$, $D$, and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\angle ABC = \angle CDE = \angle DEA = 90^\circ$. The plane of triangle $ABC$ is parallel to $\overline{DE}$. What is the area of triangle $BDE$?
3. Edge length of a cube from distances ●●●○○ 50 ptsLet $P$ be a point inside the cube $ABCD\text{-}EFGH$, where $ABCD$ and $EFGH$ are opposite faces and $E,F,G,H$ lie directly above $A,B,C,D$ respectively. Given $PA=PB=\frac{3\sqrt{3}}{2}$ and $PF=PC=\frac{\sqrt{107}}{2}$, find the edge length of the cube.
4. Center of a cube from three vertices ●●●○○ 50 ptsThree vertices of a cube in space have coordinates $A = (2,3,0)$, $B = (0,5,4)$, and $C = (4,1,8)$. Find the coordinates $(x, y, z)$ of the center of the cube.