how to substitute to poligonal forms for one?

Question

Say I have one big triangle and the other similar, and i calculate their areas and sum them having a new area how can i know what height and base the new triangle equilateral will have? all the area from the first triangle and the other must be in the new one. will its lateral sides only grow and not its base making it an iscoseles triangle? i need help on this

Answer 1

Carl,

 

Take one of the two similar triangles and flip it on its vertex and slide them together like a puzzle so the vertex of triangle 2 just touches the base of triangle 1.  Now above, the vertex of triangle 1 will touch the base of similar triangle 2.  This is so because you're told that an equilateral results , actually a slanted one with opposing sides equal in length. And since the triangles fit "snugly" together to form the equilateral, its area is just the sum of the two equal triangle areas.

 

To prove this algebraically/geometrically you can construct perpendiculars from each triangle's vertex to its base.  Cut off the smaller triangle from the left side that's outside the left perpendicular line and move it over to the right side. Again, it fits "snugly".  You now have a rectangle of height = length of the perpendicular line and width = original base of triangle 1.