4. In a triangle, two medians are perpendicular to each other, their lengths are 18 cm and 24 cm. Calculate the area of the triangle.
Solution. $S=\frac{1}{2} A C \cdot B M$ (see the figure). By the property of medians, we have $A O=\frac{2}{3} \cdot A E=16, O C=\frac{2}{3} \cdot C D=12$. From $\triangle A O C$, we find $A C=\sqrt{A E^{2}+O C^{2}}=20$. Let $D F \perp A C, O K \perp A C$,

![](https://cdn.mathpix.com/cropped/2024_05_06_b4606a2fddb2d21d9172g-1.jpg?height=388&width=446&top_left_y=2359&top_left_x=1479)
moreover, $B M \perp A C$. Note that $S_{A O C}=\frac{1}{2} A O \cdot O C=\frac{1}{2} K O \cdot A C$, hence $K O=\frac{A O \cdot O C}{A C}=9.6$. Triangles $F D C$ and $K O C$ are similar, so $\frac{D F}{K O}=\frac{D C}{O C}$, from which we get $D F=\frac{D C \cdot K O}{O C}=14.4$, and $B M=2 D F=28.8$. Therefore, we have $S=\frac{1}{2} A C \cdot B M=288$.

Answer: 288.