5. In a regular quadrilateral pyramid $P-ABCD$, the side faces are equilateral triangles with a side length of 1. $M, N$ are the midpoints of edges $AB, BC$ respectively. The distance between the skew lines $MN$ and $PC$ is $\qquad$
Answer: $\frac{\sqrt{2}}{4}$.
Solution: Let the diagonals $A C, B D$ of the base intersect at point $O$, and draw a perpendicular from point $C$ to line $M N$, intersecting $M N$ at point $H$.

Since $P O$ is the perpendicular to the base, $P O \perp C H$, and also $A C \perp C H$, so $C H$ is perpendicular to the plane $P O C$, hence $C H \perp P C$.

Therefore, $C H$ is the common perpendicular segment of lines $M N$ and $P C$, and $C H=\frac{\sqrt{2}}{2} C N=\frac{\sqrt{2}}{4}$, so the distance between the skew lines $M N$ and $P C$ is $\frac{\sqrt{2}}{4}$.