## SUBJECT 3

Let the collinear points $A, B, C, D$ in this order, such that $5 A B=9 A C-4 A D$ and $B D=18 \mathrm{~cm}$.

a) Find the lengths of the segments $B C$ and $D C$.

b) If $P$ is the midpoint of segment $A D$ and $P \in(B C)$, specify the maximum possible value, a natural number, of the length of segment AD.
## SUBJECT 3

a) If $\mathrm{AB}=\mathrm{x}, \mathrm{BC}=\mathrm{y}, \mathrm{CD}=\mathrm{z}$ we have:

$$
\begin{aligned}
& 5 x=9(x+y)-4(x+y+z) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\
& 5 x=9(x+y)-4(x+18) \Rightarrow y=8 \text { and } z=10 \Rightarrow BC=8 \text{ cm and } CD=10 \text{ cm} \ldots \ldots \ldots
\end{aligned}
$$

$2 p$
b) $\mathrm{P} \in(BC)$ and $\mathrm{B} \in(AC) \Rightarrow AP>AB$

$$
\frac{x+y+z}{2}>x \Rightarrow x<y+z
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_6ca0bbdf59da41135747g-2.jpg?height=63&width=1499&top_left_y=1839&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_6ca0bbdf59da41135747g-2.jpg?height=52&width=1299&top_left_y=1910&top_left_x=367)