2002 AMC 12P Problems/Problem 24

Revision as of 18:48, 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

Assume points $P$, $Q$, and $R$ are on faces $ABD$, $ACD$, and $BCD$ respectively such that $EP \perp ABD$, $EQ \perp ACD$, and $ER \perp BCD$.

Assume points $S$, $T$, and $U$ are on edges $AB$, $AC$, and $BC$ respectively such that $ES \perp AB$, $ET \perp AC$, and $EU \perp BC$.

Consider triangles $EPS$, $EQT$, and $ERU$. Each of these triangles have a right angle and an angle equal to the dihedral angle of the tetrahedron, so they are all similar by AA similarity.

See also

2002 AMC 12P (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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