Given: triangle ABC, median segment AD, AD=1/2 BC how do you prove Triangle ABC is a right triangle, and segment BC is its hypotenuse?

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Answer 1 · 2015-04-22T12:33:26

The median segment divides the triangle into 2 isosceles triangles and everything else is trivial.

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#|AD| = (|BC|)/2 =|BD|#
So
#Delta ADB# is isosceles with
#/_ DAB = /_DBA#

Similarly
#Delta ADC# is isosceles with
#/_DAC = /_DCA#

#/_CAB = /_CAD + /_BAD#

#= /_ACB + /_ABC#

Therefore
#/_CAB# is a right angle
and
#BC# is the hypotenuse of a right-angled triangle.