Question 1147732
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Let the triangle be ABC, with AB=14, BC=48, and CA=50.  Those side lengths are a Pythagorean Triple; the triangle is a right triangle.<br>
Let M be the midpoint of AB and let N be the midpoint of CA.  Then Triangles ABC and AMN are similar, in the ratio 2:1.  So triangle AMN has side lengths 7, 24, and 25.<br>
We are to find the length of MP, where P is the point on CA for which MP is perpendicular to CA.<br>
The area of triangle AMN, using AM and MN as the base and height, is (1/2)(7)(24) = 84.<br>
The area of the same triangle, using AN and MP as the base and height, is (1/2)(25)(MP).<br>
So<br>
{{{(1/2)(25)(MP) = 84}}}
{{{(25/2)(MP) = 84}}}
{{{MP = 84*(2/25) = 168/25}}}<br>
ANSWER: The perpendicular distance from the longest side to the midpoint of the shortest side is 168/25 cm, or 6.72cm.