<p>How would I do this? </p>
<p>"There are three points A, B, and C in a plane. The distance from A to B is 4 cm, and the distance from B to C is 3 cm. All the possible locations of point C form a plane geometric figure. What is the area of this figure in square cm?"</p>
<p>Where is this problem from? I feel like this is a badly worded problem - here’s why:</p>
<p>If we fix A and B (or just fix B), then C can be any point that is 3 cm away from B (i.e. C is on the circle with center B and radius 3). The set of possible points for C has zero area, but encloses a circular region of area 9pi. Technically the answer would be zero in this case.</p>
<p>If we fix A and let B, C vary, then 1 <= AC <= 7, and it can be shown that C can be anywhere in the plane such that 1 <= AC <= 7. The area of the set of points where C could be is 49pi - pi = 48pi.</p>
<p>Of course, we can just pick C to be anywhere on the plane, and then pick A and B such that the constraint are satisfied, in which the answer is infinity.</p>
<p>It’s not from College Board, just a problem I saw somewhere. The answer was 9 pi. I guess they wanted us to assume B is fixed and C makes a circle, and find the area of the circle. I guess that’s a good problem if it’s worded correctly.</p>
<p>I just assumed A & B were fixed and got 9 pi for the circle C made, but I didn’t realize all the other possibilities you mentioned. Good job :).</p>
<p>@cloudeleven even then, it is still not the best problem. The set of possible locations for C, given B is fixed, <em>encloses</em> a region of area 9pi, but technically the area described above should be zero. I guess that’s another wording issue.</p>
<p>I may appear nit-picky on this problem, but as a math major and someone who has written mock and actual contest problems, tiny details on wording do matter and the quality of a test is seriously diminished if there are lots of poorly-worded or ambiguous problems. Also it’s generally best to only use CB materials for SAT preparation.</p>