Alice was supposed to find a cross-section of a pyramid. The task stated:
"Given a pyramid $ABCDV$ with a square base, find the cross-section of the pyramid by the plane $KLM$, where point $K$ lies on edge $AB$, point $L$ lies on edge $BC$, and point $M$ lies on edge $CV$ (see the picture)."
Alice proceeded as follows:
(1) She connected the points $K$ and $L$ and claimed that the line segment $KL$ is one side of the cross-section.
(2) She connected the points $L$ and $M$ and claimed that the line segment $LM$ is another side of the cross-section.
(3) She drew the ray $KL$ and the ray $DC$. By intersecting these rays, she obtained an auxiliary point $E$. She claimed that this point $E$ belongs to the plane $KLM$.
(4) She drew the line $EM$ and marked its intersection with edge $DV$ as point $F$. She claimed that the line segment $FM$ is the next side of the cross-section.
(5) She drew the line $p$ parallel to line segment $LM$, passing through the point $F$, and marked its intersection with edge $AD$ as the point $H$. She claimed that the line segment $FH$ is also a side of the cross-section. Then she marked the pentagon $KLMFH$ as the desired cross-section.
Did she make any mistake? If so, identify in which step and explain.
Yes, she made a mistake in step (1). The line segment $KL$ is not a side of the sought cross-section.
Yes, she made a mistake in step (2). The line segment $LM$ is not a side of the sought cross-section.
Yes, she made a mistake in step (3). The point $E$ does not belong to the plane $KLM$.
Yes, she made a mistake in step (4). The line segment $FM$ is not a side of the sought cross-section.
Yes, she made a mistake in step (5). The line segment $FH$ is not a side of the sought cross-section.
Alice made a mistake in step (5) because the line $p$ is skew to the line $AD$. Alice should have proceeded as follows:
- Draw the line $KL$ and the ray $DC$. By intersecting them, obtain an auxiliary point $E$.
- Connect point $E$ with point $M$ and mark the intersection of the line $EM$ with edge $DV$ as point $F$.
- Extend the line segment $DA$ into the ray $DA$ and mark its intersection with the line $KL$ as point $H$.
- Construct the line $HF$ and mark its intersection with edge $AV$ as point $J$.
- Declare the pentagon $KLMFJ$ as the desired cross-section.
