No, the quotient of two perfect squares is not always a perfect square. take 16/9 for example, it equals 1.77777777(bar notation), which is most certainly not a perfect square.
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If the area of the smaller square is 256, then its side length is 16. Putting 16 into the ratio, the new ration would be 16:36 because both the 4 and 9 are multiplied by 4. So, the side length of the larger triangle is 36.