[T doubling the median]

In an isosceles triangle with a lateral side equal to 4, a median is drawn to the lateral side. Find the base of the triangle if the median is equal to 3.

#
Complete the given triangle to a parallelogram.

## Solution

Let \( x \) be the base \( BC \) of the isosceles triangle \( ABC \). On the extension of the median \( BM \) beyond point \( M \), we lay off the segment \( DM \) equal to \( BM \). Then \( BADC \) is a parallelogram. Therefore,

\[
AC^2 + BD^2 = 2(AB^2 + BC^2), \text{ or } 16 + 36 = 2 \cdot 16 + 2x^2.
\]

From this, we find that \( x^2 = 10 \).

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-31.jpg?height=332&width=672&top_left_y=1792&top_left_x=699)

Answer

\(\sqrt{10}\).