Sophie was tasked with constructing any trapezoid based on the following instructions:
„Construct a trapezoid $ABCD$, where $AB$ and $CD$ are bases, and it is given that $|AB| = 6\,\mathrm{cm}$, $|CD| = 1.5\,\mathrm{cm}$, $|AD| = 4\,\mathrm{cm}$, $|BC| = 5\,\mathrm{cm}$."
Sophie proceeded as follows (see the picture below):
(1) Sophie constructed the line segment $AB$ of length $6\,\mathrm{cm}$.
(2) She drew a circle $d$ centered at point $A$ with a radius of $4\,\mathrm{cm}$. She claimed that point $D$ lies on circle $d$.
(3) She drew a circle $c$ centered at point $B$ with a radius of $5\,\mathrm{cm}$. She claimed that point $C$ lies on circle $c$.
(4) She chose point $D$ arbitrarily on circle $d$.
(5) She realized that, knowing the position of point $D$ of the desired trapezoid, then point $C$ must lie on a circle $e$ centered at point $D$ with a radius of $1.5\,\mathrm{cm}$. Then she realized that point $C$ must lie at the intersection of circles $c$ and $e$.
(6) She constructed the trapezoid $ABCD$.
However, after Sophie analyzed her picture, she found that the constructed object is just a quadrilateral, not a trapezoid. Where did she make a mistake?
She made a mistake in step (1).
She made a mistake in step (2).
She made a mistake in step (3).
She made a mistake in step (4).
She made a mistake in step (5).
Sophie didn't construct the trapezoid correctly. She made a mistake in step (4). She can't choose the position of vertex $D$ arbitrarily. She did not consider that $AB$ and $CD$ must be parallel. She could have proceeded as follows:
- Construct the line segment $AB$, and on it mark a point $A_1$, such that $|A_1B|=|CD|= 1.5\,\mathrm{cm}$.
- Draw a circle $c$ centered at point $A$ with a radius of $|AD|=4\,\mathrm{cm}$.
- Draw a circle $d$ centered at point $A_1$ with a radius of $|BC|=5\,\mathrm{cm}$.
- The intersection of circles $c$ and $d$ gives the vertex $D$ of the trapezoid.
- Translate the line segment $A_1D$ by the vector $\overrightarrow{A_1B}$. This translation gives the line segment $BC$.
This procedure provides all vertices of the desired trapezoid $ABCD$ (see the picture below).
