3. In parallelogram $A B C D$, the internal angle $\alpha$ is acute, side $B C$ measures $4.5 \mathrm{~cm}$. On side $A B$ lies point $G$ such that $|A G|:|G B|=2: 3$. The line through $C G$ intersects the line through $A D$ at point $P$. Calculate the length of segment $A P$. Draw a sketch.
3. We draw the appropriate sketch. We determine that triangles $A G P$ and $G B C$ are similar. We express the lengths $|A G|=2 x$ and $|G B|=3 x$. We consider that the ratio of corresponding sides is constant: $|A G|:|G B|=|A P|:|B C|$, express and calculate $|A P|=\frac{|B C| \cdot|A G|}{|G B|}=\frac{4.5 \cdot 2 x}{3 x}=3$.

Sketch of the parallelogram with the correct input $G 1$ point Observation or consideration that triangles $A G P$ and $G B C$ are similar. . 1 point Writing or considering $|A G|=2 x$ and $|G B|=3 x$. 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-09.jpg?height=577&width=737&top_left_y=1790&top_left_x=1139)
Writing the ratio $|A G|:|G B|=|A P|:|B C|$ ..... 1 point
Expressing $|A P|=\frac{|B C| \cdot|A G|}{|G B|}$. ..... 1 point ..... 1 point
Calculation $\frac{4.5 \cdot 2 x}{3 x}=3$