bikerguy.gmat wrote:Stuart Kovinsky wrote:The shortest distance between any two points is a straight line. Therefore, every side of a triangle has to be LESS than the sum of the other two sides.
In other words, going straight from point A to point B MUST be shorter than going from A to C and then C to B.
For this rule to hold true for all 3 sides of a triangle, every side must also be GREATER than the difference between the other two sides.
So, the general rule is:
|Side 2 - Side 3| < Side 1 < Side 2 + Side 3
and this rule holds true for every side.
Hi Stuart, I do not understand this. Can you please explain in more detail.
Never heard about, or seen this kind of rule anywhere else. And, I could not do it using geometry principles too.
Thanks in advance!
Sure!
Let's start with two houses, A and B.
Let's build a road in a straight line directly from A to B. That's the shortest possible road that we can build to connect the two houses.
Now let's build a third house, C, and build direct roads from A to C and from C to B.
We now have two possible routes to get from A to B. We can go directly from A to B, or we can go from A to C to B.
Another way we can think of AC and CB is as a detour from points A to B. Since the shortest distance between A and B is our original straight road, we can see that:
AB < AC + CB
Now for the second part of the relationship. What we just determined about AB must also be true for AC and CB. In other words:
AC < AB + CB
and
CB < AB + AC
We simply rearrange these inequalities:
AB < AC + CB
provides:
AB - CB < AC
AB - AC < CB
AC < AB + CB
provides:
AC - AB < CB
AC - CB < AC
CB < AB + AC
provides:
CB - AB < AC
CB - AC < AB
Putting those 9 inequalities together, we get:
|AB - AC| < CB < AB + AC
|AC - CB| < AB < AC + CB
|CB - AB| < AC < CB + AB
