1. First, we notice that the cosine of \(\angle ABC = 4/5\)(adjacent / hypotenuse).
With this, we can use the law of cosines to find the length of the median \(\overline {AM}\).
We get the equation:
\(\overline {AM}^2 = 4^2 + (2.5)^2 - 2*4*2.5* \cos(\angle ABC)\)
We combine like terms and get:
\(\overline {AM}^2 = 22.25 - 20* \cos(\angle ABC)\)
We realize we already have \(\cos( \angle ABC)\)= \(\frac45\)
We then substitute, and get:
\(\overline{AM}^2 = 22.25-20*\frac45 = 22.25 - 16 = 6.25\)
To get the length of the median \(\overline {AM}\), we just take the square root of this result, to get:
\(\overline{AM} = \sqrt{6.25} = 2.5\)
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