MathDB

A unit cube has vertices

P_1, P_2, P_3, P_4, P_1', P_2', P_3'

, and

P_4'

. Vertices

P_2, P_3

, and

P_4

are adjacent to

P_1

, and for

1\leq i\leq 4

, vertices

P_i

and

P_i'

are opposite to each other. A regular octahedron has one vertex in each of the segments

P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'

, and

P_1'P_4'

. What is the octahedron's side length? [asy] import three;
size(7.5cm); triple eye = (-4, -8, 3); currentprojection = perspective(eye);
triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};
// draw octahedron triple pt(int k){ return (3*P[k] + P[1])/4; } triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; } draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6)); draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6)); draw(ptp(2)--pt(4), gray(0.6)); draw(pt(2)--ptp(4), gray(0.6)); draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4")); draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));
// draw cube for(int i = 0; i < 4; ++i){ draw(P[1]--P); draw(Pp[1]--Pp); for(int j = 0; j < 4; ++j){ if(i == 1 || j == 1 || i == j) continue; draw(P--Pp[j]); draw(Pp--P[j]); } dot(P); dot(Pp); dot(pt(i)); dot(ptp(i)); }
label("

P_1

", P[1], dir(P[1])); label("

P_2

", P[2], dir(P[2])); label("

P_3

", P[3], dir(-45)); label("

P_4

", P[4], dir(P[4])); label("

P'_1

", Pp[1], dir(Pp[1])); label("

P'_2

", Pp[2], dir(Pp[2])); label("

P'_3

", Pp[3], dir(-100)); label("

P'_4

", Pp[4], dir(Pp[4])); [/asy]

<span class='latex-bold'>(A)</span>\ \frac{3\sqrt{2}}{4}\qquad<span class='latex-bold'>(B)</span>\ \frac{7\sqrt{6}}{16}\qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt{5}}{2}\qquad<span class='latex-bold'>(D)</span>\ \frac{2\sqrt{3}}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt{6}}{2}

Problem Statement