Question 1066322
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1.  Make a sketch.
    Let ABC be your right-angled triangle with the right angle vertex A.
    Let BD and CE be these two medians (|CE| = 2*sqrt(13) and |BD| = sqrt(73), for certainty).
    Let x and y be the lengths of the legs AB and AC respectively.


2.  From the right triangle ACE and ABD you have

    {{{abs(AC)^2 + abs(AE)^2}}} = {{{abs(CE)^2}}}   (1)

    {{{abs(AB)^2 + abs(AD)^2}}} = {{{abs(BD)^2}}}   (2)

or

    {{{(x/2)^2 + y^2}}} = {{{(2*sqrt(13))^2}}}     (3)

    {{{x^2 + (y/2)^2}}} = {{{(sqrt(73))^2}}},      (4)

By adding equations (3) and (4), you get

    {{{(5/4)*x^2 + (5/4)*y^2}}} = 4*13 + 73 = 125,   or

    {{{x^2 + y^2}}} = 100.


3.  Thus {{{x^2 + y^2}}} = 100.

    It is the length of the hypotenuse squared, {{{abs(BC)^2}}}.

    Hence, the length of the hypotenuse {{{abs(BC)}}} = {{{sqrt(100)}}} = 10 cm.
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Solved.  &nbsp;&nbsp;Answer</U>: &nbsp;&nbsp;The length of the hypotenuse is 10 cm.