Since the angles are all the same, the two triangles are equivalent. Therefore, both the base and the height necessarily increased by the same proportion (which is what we are trying to find). Remember, the area of a triangle is 1/2*b*h.

The new b and h, when multiplied together, should give an answer twice as great as before. Since the proportions are equal, what number, when multiplied by itself, yields an answer of 2? The answer is, of course, the square root of 2.

Mathematically, if the proportion is labeled x,

2*(1/2*b*h) = 1/2*xb*xh

2bh = xbxh

2bh = x^2bh

2 = x^2

root(2) = x