In the figure above, PQRT is a rectangle. What is the length of segmen

We can do everything algebraically: If we use W and L to represent the width and length of the rectangle, so W is the length of PQ (which we want to find), and if we use B to represent the line TS, which is the base of the triangle, then the area of the rectangle is LW, and the area of the triangle is BW/2.

Statement 1 tells us that if we add the rectangle's area and the triangle's area, we get 39. It also tells us that B = 6. So

LW + (BW/2) = 39
LW + (6W/2) = 39
LW + 3W = 39
W(L+3) = 39

So we might have W = 1 and L = 36, say, or W = 3 and L = 10, among other possibilities.

But there's a much faster way to see that Statement 1 is not sufficient. If you just draw the triangle's base TS of length 6, and draw some height RT that isn't too long (so the area of the triangle alone is less than 39), if we then start extending a rectangle to the left by drawing PT and QR, the area of the entire shape will get larger and larger the longer we make the rectangle's length. For exactly one length of the rectangle, the area of the entire shape will be exactly 39. But that length will be different for any height we choose - the length will need to be big if the height is small, and will need to be small if the height is big, so we can't possibly find the dimensions of the figure.

Statement 2 tells us LW = 30 and L = 10, so W = 3 and Statement 2 is sufficient. The answer is B.