Figure Not Drawn to ScaleDate: 08/16/2003 at 16:52:31 From: Jeff Subject: Figure not drawn to scale A line l is drawn with points A,B,C,D,E, and F, in that order. The points appear equidistant, but the figure is not drawn to scale. If AD=BE in the figure, then which of the following must be true? A. AB=EF B. AC=CE C. AB<DF D. AC<CF E. BC<CE What can I assume without being mistaken? Date: 08/16/2003 at 23:01:34 From: Doctor Peterson Subject: Re: Figure not drawn to scale Hi, Jeff. You can only assume what they've explicitly told you: that AD=BE, and that the five points are in the indicated order on the line. ---A-----B------C-----D-------E----F--- |<--------a------->| | |<---------a-------->| The only implication (in terms of equality) that you can draw from this is that AB = DE. Do you see why that is true? We know nothing about how far away F is, so (A) is certainly unknown. And we don't know where C is within BD, so (B) is out. The inequalities will have to be deduced primarily from the order, possibly using the one equality we know. (3) AB < DF? We know that AB = DE; and the fact that F is outside of DE tells us that DE < DF. So AB = DE < DF and this inequality is true. (4) AC < CF? Since we don't know just where C and F are, the best we can do is to consider the extremes. If C is right next to B, and F is far out, then AC < CF; but if C is right next to D and F is right next to E, then AC will be considerably greater than AB while CF will be close to DE, which is equal to AB, so AC > CF. So we can't tell whether this one is true. (5) BC < CE? If C is close to D, and DE is small, then this would be false, though as drawn it is true. That solves the problem. Looking back, what did I use to do it? To prove that (3) is true, I used deduction from the facts I had. Given that only one could be true, that's enough to have done. But to prove that (4) and (5) are unknown (which I might have had to do if I had more trouble proving (3)), I had to look for counterexamples: cases in which they are false, which I found by thinking about extreme cases. If the problem had been to show which statement(s) are either known to be true or known to be false, I would have had to do all the work I did here. Your actual problem was a lot easier; but it's a good habit to play with problems like this and go beyond what was asked, in order to build your deductive skills. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/17/2003 at 08:34:59 From: Jeff Subject: Thank you (Figure not drawn to scale) Thank you, Dr. Peterson. |
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