Find the area of a triangle if two of its sides are equal to 1 and $\sqrt{13}$, and the median drawn to the third side is equal to 2.

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Using the formula for the median of a triangle, find the third side of the triangle.

## Solution

Let in triangle $ABC$, $AB=\sqrt{13}$, $AC=1$, and the median $AM=2$. From the formula for the median of a triangle (see problem $\underline{\text{5L}} 5267$), it follows that

$4 AM^{2}=2 AC^{2}+2 AB^{2}-BC^{2}$, which means $BC^{2}=2+26-16=12$.

Since $AB^{2}=AC^{2}+CB^{2}$, triangle $ACB$ is a right triangle, and $S_{ACB}=1/2 \cdot AC \cdot BC=\sqrt{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_96d74c2697cd905bd948g-03.jpg?height=269&width=792&top_left_y=0&top_left_x=639)

## Answer

$\sqrt{3}$.