Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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.
```
Associated Topics:
High School Euclidean/Plane Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search