Show that if a median of a triangle is one-half the side to which it is drawn, then it must be right.

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- Dec 22nd 2009, 07:20 AMsologuitarMedian of a Triangle
Show that if a median of a triangle is one-half the side to which it is drawn, then it must be right.

- Dec 22nd 2009, 08:03 AMGrandad
Hello sologuitarDo you know the theorem "The angle in a semicircle is a right-angle"? If you do, use this, noting that the three vertices of the triangle are equidistant from the mid-point of the side to which the median is drawn. Thus, this point is the centre of the circle passing through the vertices of the triangle - and using the above theorem, the result follows.

If you don't know this circle theorem, then use the fact that you have two isosceles triangles - one on either side of the median. Then look at the angles in these triangles that are equal. Call the size of one pair $\displaystyle x$, and the size of the other pair $\displaystyle y$. Then $\displaystyle 2x+2y = 180^o$ (can you see why?). Therefore $\displaystyle x+y=90^o$.

Can you fill in the details now?

Grandad