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The median to the hypotenuse
Pythagorean Theorem (direct and inverse)
Mean proportionals in a right triangle

In a right triangle $A B C\left(\angle C=90^{\circ}\right)$, the height $C D$ and the median $C E$ are drawn. The areas of triangles $A B C$ and $C D E$ are 10 and 3, respectively. Find $A B$.
$D E: A B=S_{C D E}: S_{A B C}$.

## Solution

Notice that $D E: A B=S_{C D E}: S_{A B C}=3: 10$. Therefore, $B C^{2}: A C^{2}=B D: A D=(5+3):(5-3)=4: 1$, which means $B C = 2 A C$.

Thus, $A C^{2}=S_{A B C}=10, A B^{2}=5 A C^{2}=50$.

![](https://cdn.mathpix.com/cropped/2024_05_06_a7c46f4bbe36942e4ae1g-32.jpg?height=260&width=440&top_left_y=2517&top_left_x=818)

## Answer

[The ratio of the areas of triangles with a common base or a common height]

Problem