2002 AMC 12P Problems/Problem 24

Revision as of 17:25, 10 March 2024 by The 76923th (talk | contribs) (Solution)

Problem

Let $ABCD$ be a regular tetrahedron and Let $E$ be a point inside the face $ABC.$ Denote by $s$ the sum of the distances from $E$ to the faces $DAB, DBC, DCA,$ and by $S$ the sum of the distances from $E$ to the edges $AB, BC, CA.$ Then $\frac{s}{S}$ equals

$\text{(A) }\sqrt{2} \qquad \text{(B) }\frac{2 \sqrt{2}}{3} \qquad \text{(C) }\frac{\sqrt{6}}{2} \qquad \text{(D) }2 \qquad \text{(E) }3$

Solution

We create a coordinate system for the point $E = (x, y)$. Let $x$ be the distance from $AB$, and let $y$ be the distance $EF$, where $F$ is on $AC$ and $EF$ is parallel to $AB$. Call the distances from $E$ to $ABD$, $ACD$, and $BCD$ $d_a$, $d_b$, and $d_c$, respectively.

See also

2002 AMC 12P (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AMC 12 Problems and Solutions

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