| You want to find the range of possible area values. Any answer choice(s) that isn't within this range is not a possible area.
First, find the minimum possible area. You know that the third side must be greater than one unit long. If you think about it, the smallest possible length is not an integer. It would be 1.000000000(infinite number of 0's)1. If you try to picture this 7-6-1.000001 triangle in your head, you can figure out that the area would approach 0.
The maximum possible area of any triangle is when it's a right triangle. if you didn't already know this, you can figure it out using the formula A=BH/2. Take a right triangle whose legs' lengths are b and h. The area is bh/2. Now, if you tilt h over a little bit to make the triangle acute, the base stays the same (b) while the height decreases (a fraction of h). Now the area is (a part of x)(b)/2. This area is smaller than bh/2. The same thing happens if you tilt h to make the triangle obtuse.
Now, if the triangle is right, then the legs would have lengths 6 and 7. The area is 6*7/2, or 21.
The range of the area is 1 < A <= 21.
Choices I and II are within this range, but III is not, so the answer is D. |