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3 problems match “centroid”. Clear

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1. Centroid of a triangle ●○○○○ 10 ptsFind the centroid of the triangle with vertices $A(4, 5)$, $B(20, 25)$ and $C(30, 6)$. Give the $x$- and $y$-coordinates of the centroid.
2. Centroid distance to a plane ●●●○○ 50 ptsThe vertices $A$, $B$, $C$ of an equilateral triangle $ABC$ with side length $6$ are at distances $1$, $2$ and $3$ from a plane $\alpha$. Find all possible distances from the centroid $G$ of triangle $ABC$ to the plane $\alpha$. Round each value to two decimal places and give them in increasing order.
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.