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### Midpoint Formula for any Fraction

```
Date: 01/26/99 at 00:04:23
From: T.J.
Subject: Rectangular coordinate system

I need to come up with a formula similar to the midpoint formula. I
need this formula to find "any fractional part" of the line from P1
to P2.

This is the midpoint formula:

X1 + X2             Y1 + Y2
Xm = -------  and  Ym =  -------
2                   2

The formula I came up with is:

X1 + X2            Y1 + Y2
Xa = -------  and  Ya = -------
N                  N

where N represents any fraction of the line and Xa is the point on the
line. Will this formula actually work?

This is where I am stuck.
Thanks!
```

```
Date: 01/26/99 at 12:05:52
From: Doctor Peterson
Subject: Re: Rectangular coordinate system

Hi, T.J. Let's start out in one dimension to make it easier. We have
two points A and B, and want to find a point C that is F = m/n of the
way from one to the other. For instance, F might be 2/3. Here's what
we have:

0  A                     C      B
+--+---------------------+------+----
|<-------------------------->|
|             B-A     |      |
|<------------------->|      |
|       F*(B-A)       |      |

The coordinate of point C is

A + F*(B-A) = A + F*B - F*A = (1-F)*A + F*B)

So the point you are looking for in your two-dimensional problem is

Xf = (1-F)*X1 + F*X2
Yf = (1-F)*Y1 + F*Y2

If you have ever used weighted averages, you'll see that that is just
what this formula is. The midpoint is a simple average (mean), while
your fractional point is the weighted average of the endpoints.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
Middle School Geometry

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