A museum exhibit, ABCD, has infrared beams around it for security, as shown below.
If the length of the beams is made three times the original length on each side, which statement is correct about the maximum area available for an exhibit to be displayed?
It becomes nine times the original area.
It becomes twenty-seven times the original area.
It becomes three times the original area.
It becomes eighty-one times the original area.
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Imagine x,y,z to be the lengths of the sides where they are labeled. I multiplied each portion by 6 to get 6x, 6y, and 6z. From there, we find the new size of each area, which is 36xy + 18xz. The old area was just xy + xz. So what is the difference in Areas?
The parallelogram here is a trapezoid. Thus the area of this parallelogram is:
. In this case, b1 = AB and b2 = CD. and the height (h) is AD. When each length is timed by six, the new sides are just 6 times the original length of each. Plugging this in, the new area is:
. Since the original area is 1/2(b1+b2)h, the new area is just 36/3 times the original area