“Consider the cube $ABCDEFGH$ with points $I$ and $J$. Point $I$ lies on the half-line $FE$, and point $J$ is the midpoint of edge $FG$ (see the picture). Construct the cross-section of the cube by the plane $IJB$.”
Lucas drew the cube, marked the given points $I$ and $J$, and proceeded as follows:
(1) First, he connected points $I$ and $J$ and marked the intersection of the line segment $IJ$ with edge $EH$ as point $K$. He claimed that the line segment $JK$ is one of the sides of the cross-section.
(2) Next, he connected points $K$ and $A$ and claimed that the line segment $KA$ is another side of the cross-section
(3) Then, he connected points $J$ and $B$ and claimed that the line segment $JB$ is the third side of the cross-section.
(4) He constructed the quadrilateral $ABJK$ as the sought cross-section.
Did Lucas make any mistakes? If so, 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 $KA$ is not a side of the sought cross-section.
Yes, he made a mistake in step (3). The line segment $JB$ is not a side of the sought cross-section.
No, he did not make any mistakes.
The mistake is in step (2). The line segment $KA$ is not a side of the required cross-section because point $A$ does not lie in the plane $IJB$.
The correct procedure is as follows:
(1) Connect the points $I$ and $J$, and mark the intersection of the line segments $IJ$ with edge $EH$ as point $K$.
(2) Connect points $J$ and $B$.
(3) Construct a line parallel to the line segment $JB$ that passes through point $K$, and mark its intersection with edge $AE$ as point $L$.
(4) Connect points $L$ and $B$.
(5) The quadrilateral $LBJK$ is the sought cross-section.
