computational geo with a tetrahedron 2010 PUMaC Team Round 1 Across

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High School Math Grades 9-12, Ages 13-18

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computational geo with a tetrahedron 2010 PUMaC Team Round 1 Across

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#1 h Dec 26, 2021, 5:56 PM

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Tetrahedron $ABCD$ has base $\triangle BCD$ which is an equilateral triangle with side length $2$ . $AB = AC = 3$ and $AD = 4$ . Let $M$ be the centroid of tetrahedron $ABCD$ and $N$ be the centroid of $\triangle BCD$ . Write $\displaystyle\frac{AN}{AM} = \displaystyle\frac{p}q$ , where $p$ and $q$ are relatively prime integers. You are to submit the concatenation of $p$ and $q$ .

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#2 h Feb 22, 2024, 8:51 PM

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Given the tetrahedron $(A,BCD)\ :\ B(0,-1,0),C(\sqrt{3},0,0),D(0,1,0),A(\frac{11}{4\sqrt{3}},-\frac{7}{4},\frac{\sqrt{71}}{2\sqrt{3}})$ .

Points $N(\frac{\sqrt{3}}{3},0,0)$ and $M(\frac{23}{16\sqrt{3}},-\frac{7}{16},\frac{\sqrt{71}}{8\sqrt{3}})$ .

Distances $AN=\sqrt{10}$ and $AM=\frac{3\sqrt{10}}{4}$ .

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