A very effective way to solve Quantitative Comparison Questions is to: 1) Plug in values and 2) Think of extreme cases.
So let's plan to: 1) Find an example of a triangle where we can easily calculate its area and perimeter, 2) Think of an extreme case that maximizes the area of a quadrilateral with the same perimeter as the right triangle, and 3) Think of an extreme case that minimizes the area of a quadrilateral with the same perimeter as the right triangle. The goal is to find the largest and smallest values for Quantity B while holding Quantity A constant.
Right triangles have "Pythagorean Triples," a fancy way of saying that all of the sides are integers. The most common Pythagorean Triple is a 3:4:5 triangle, so let's use that for Quantity A. The perimeter would be 3+4+5=12, and the area would be 0.5*3*4 = 6. So now let's think of a quadrilateral with the largest area we can imagine that has a perimeter of 12.
Let's try a square. A square with perimeter 12 would have a side of 12/4, or 3. And the area of that square would be 3*3, or 9. So in this case, Quantity B is greater, so we can eliminate answer choices A and C.
Next, let's try to minimize the area of a quadrilateral. Since the area of a rectangle is the product of its length and width, we can get a really small area of a rectangle for a constant perimeter if we minimize one of the sides. So for example, if we make the length 5.9 and the width 0.1, then the perimeter would be 12 and the area would be 5.9*0.1, or 0.59. So in this case Quantity A would be greater, thus eliminating Quantity B.
And the only answer choice remaining is D.
So always remember to: 1) Plug in values and more importantly, 2) Think of extreme cases.
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Corey
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