3. Parallelogram $A B C D$ is divided by the lengths $\overline{L G}, \overline{K H}, \overline{E J} \text{ and } \overline{F I}$ into parallelograms (see figure). The perimeter of parallelogram $K R J D$ is $32 \text{ cm}$, the perimeter of parallelogram $E F N M$ is $5 \text{ dm}$, and the perimeter of parallelogram $A B C D$ is $1 \text{ m}$. Calculate the perimeter of parallelogram $N G H P$.

![](https://cdn.mathpix.com/cropped/2024_06_03_08cce83e867c11738ef9g-6.jpg?height=445&width=1082&top_left_y=1788&top_left_x=493)
First solution.

Let $|K D|=a,|L K|=b,|A L|=c,|A E|=x,|E F|=y,|F B|=z$.

![](https://cdn.mathpix.com/cropped/2024_05_30_81794be492b317694280g-27.jpg?height=392&width=991&top_left_y=358&top_left_x=538)

With the introduced notation, the perimeter of parallelogram $N G H P$ is $2 b+2 z$.

If we express the perimeters of parallelograms $K R J D, E F N M$ and $A B C D$ in centimeters, then

$2 a+2 x=32$

$2 c+2 y=50$

$2 a+2 b+2 c+2 x+2 y+2 z=100$

Note 1: Each correctly written expression that indicates the perimeter of the corresponding parallelogram is awarded 1 POINT.

By adding the first two equations and substituting into the third, we get

$32+50+2 b+2 z=100$

$1 \text{ POINT}$

$2 b+2 z=18$

The perimeter of parallelogram $N G H P$ is $18 \text{ cm}$.

1 POINT

Note 2: Points are not deducted for the lack of a unit of measurement.

TOTAL 6 POINTS