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3 problems match “coordinate geometry”. Clear

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1. Point on the quadrant bisector ●○○○○ 10 ptsPoint $P(a+3,\ 7+a)$ lies on the bisector of the second and fourth quadrants. Find $a$.
2. 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.
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.