The areas of two equilateral triangles are 27 yd and 75 yd find the ratio of their perimeters

This is a good one. Instead of working with b*h/2, we should work with the trig formula. we know an equilateral triangle has angles of 60 degrees.

Area of a tringle = 1/2 * ab sine C, where a and b are the lengths of the two sides creating the angle C.

So if the area of the first one:

27 = 1/2 ab sine 60

The second one:

75 = 1/2 xy sine 60

we know the sides are equal so we might as well say a = b; and x = y; thus

27 = 2a sine 60

75 = 2x sine 60

And we can work out sine 60. It's now just a matter of making a and x the subject of the formulas and creating the ratio a/x. Is this clear?