4. On the ray [ $O X$ consider the points $A, B, C, D, E, F$ such that $A \in(O B)$, $B \in(A C)$, $C \in(B D)$, $D \in(C E)$, $E \in(D F)$, and $O A=2 \text{ cm}$, $A B=6 \text{ cm}$, $B C=10 \text{ cm}$, $C D=14 \text{ cm}$, $D E=18 \text{ cm}$, $E F=22 \text{ cm}$.

a) If $N$ is a point on the ray $[O X$ and $O N=42 \text{ cm}$, show that $N \in(D E)$.

b) Determine the length of the segment $[M P]$, where $M$ is the midpoint of the segment $[A B]$ and $P$ is the midpoint of the segment $[B F]$.

Note. All subjects are mandatory. Time for work: 2 hours. Each subject is graded from 0 to 7 points.

## Grading Scale for 6th Grade

Note. For any correct solution different from the one in the grading scale, the maximum score from the grading scale for the respective statement is awarded.


4. a) $O D=O A+A B+B C+C D=32 \mathrm{~cm}, O E=O D+D E=50 \mathrm{~cm}$, so $O D<O N<O E$ and $N \in(D E)$
b) $O B=8 \mathrm{~cm}, O M=\frac{O A+O B}{2}=5 \mathrm{~cm}, O E=50 \mathrm{~cm}, O P=\frac{O B+O E}{2}=40 \mathrm{~cm}$, so $M P=O P-O M=35 \mathrm{~cm}$. $3 p$

Geometric Figures

Sub. 3

![](https://cdn.mathpix.com/cropped/2024_06_07_81224d613584b479d405g-3.jpg?height=649&width=1056&top_left_y=635&top_left_x=497)

Sub. 4

![](https://cdn.mathpix.com/cropped/2024_06_07_81224d613584b479d405g-3.jpg?height=152&width=1482&top_left_y=1563&top_left_x=253)