2. Construct a trapezoid, given the lengths of its diagonals $9 \mathrm{~cm}$ and $12 \mathrm{~cm}$, the length of the midline $8 \mathrm{~cm}$, and the distance between the midpoints of the diagonals $2 \mathrm{~cm}$.
SOLUTION. Let's choose the notation according to Fig. $1, K P$ is the midline in triangle $A C D$,

![](https://cdn.mathpix.com/cropped/2024_04_17_230102406b0d66efe7a7g-2.jpg?height=404&width=764&top_left_y=896&top_left_x=651)

Fig. 1

thus $|K P|=\frac{1}{2}|D C|$, similarly $|Q L|=\frac{1}{2}|D C|,|P L|=\frac{1}{2}|A B|$, so $|P Q|=\frac{1}{2}(a-c)=$ $=2 \text{ cm}$. Since $|K L|=\frac{1}{2}(a+c)=8 \text{ cm}$, we have $a=10 \text{ cm}, c=6 \text{ cm}$. First, we construct triangle $A E C$ according to the $s s s$ theorem, then on segment $A E$ we place point $B$, through which we draw a line parallel to $C E$. This line intersects the line drawn through point $C$ parallel to $A E$ at point $D$.

Solvers should recall the concept of the midline of a trapezoid, whose length is equal to the arithmetic mean of the lengths of the bases.

AUXILIARY PROBLEMS: