Ivan was given the following task: “Consider a cube $ABCDEFGH$ with points $I$, $J$ and $K$. Point $I$ is the midpoint of edge $AD$, point $J$ is the midpoint of edge $FG$, and point $K$ is the midpoint of edge $GH$. Construct the cross-section of the cube by the plane $IJK$.”
Ivan drew the cube, marked the given points $I$, $J$, and $K$, and proceeded as follows:
(1) Ivan connected points $J$ and $K$ and stated that line segment $JK$ is one of the sides of the cross-section.
(2) He constructed a line parallel to line segment $JK$, passing through point $I$. He marked the intersection of this line with edge $AB$ as point $N$. Then he claimed that line segment $IN$ is another side of the sought cross-section.
(3) He connected points $I$, $N$, $J$, and $K$ to form a quadrilateral, claiming it to be the sought cross-section.
Did Ivan make any mistakes? If yes, identify in which step and explain.
Yes, he made a mistake in step (1). The line segment $JK$ is not a side of the sought cross-section.
Yes, he made a mistake in step (2). The line segment $IN$ is not a side of the sought cross-section.
Yes, he made a mistake in step (3). The quadrilateral $INJK$ is not the sought cross-section.
No, Ivan’s entire procedure is correct.
Steps (1) and (2) are correct. However, the obtained quadrilateral is not the sought cross-section. The correct procedure is as follows:
(3) Extend edge $CB$ and mark the intersection of the half-line $CB$ with the line $IN$ as point $L$.
(4) Connect points $J$ and $L$. Then, mark the intersection of edge $BF$ with the line segment $JL$ as point $M$.
(5) Construct a line $r$ parallel to the line $MJ$, passing through point $I$, and mark the intersection of the line $r$ with edge $DH$ as point $O$.
(6) Finally, the hexagon $INMJKO$ is the sought cross-section.
