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### Midpoint Formula and Trisection Points

```Date: 09/05/2003 at 00:40:29
From: Sarah
Subject: The Cartesian Plane and the Distance Formula

Show that

((1/3)(2x_1+ x_2), (1/3)(2y_1+y_2))

is one of the points of trisection of the line segment joining
(x_1,y_1) and (x_2,y_2).

It seems to me that the answer should be related to the midpoint
formula, but I just can't figure it out.
```

```
Date: 09/05/2003 at 12:52:42
From: Doctor Peterson
Subject: Re: The  Cartesian Plane and the Distance Formula

Hi, Sarah.

You're right, it is related to the mispoint formula; if you
understand where that comes from, you should be able to derive this
formula. So they are testing your deeper understanding of the idea of
the midpoint.

Here is our segment:

(x_2,y_2)
o
/|
/ |
/  |
/   |
/    |
o-----+
/|     |
/ |     |
/  |     |
/   |     |
/    |     |
o-----+-----+
/|     |     |
/ |     |     |
/  |     |     |
/   |     |     |
/    |     |     |
o-----+-----+-----+
(x_1,y_1)

Note that when we divide the segment into three equal parts, we are
also dividing the horizontal and vertical legs of this right triangle
into three equal parts. (Think similar triangles!)

Now, in terms of the coordinates, how long is that horizontal
segment? How long is each third? How much do we have to add to x_1 to
find the x-coordinate of each intermediate point? What are those
coordinates?

I think that should lead you through the work. If you have any
further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 09/09/2003 at 23:14:19
From: Sarah
Subject: Thank you

Thank you very much!  It was much simpler than I thought.
```
Associated Topics:
High School Coordinate Plane Geometry
High School Euclidean/Plane Geometry

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