Figure Not Drawn to Scale

Date: 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.