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### Using Midpoints to Determine Vertices

```Date: 09/04/2002 at 21:01:01
From: Leanne
Subject: Using midpoints to determine vertices

We have been learning how to calculate the length of line segments,
and one of our questions read like this:

The midpoints of the sides of a triangle have coordinates G(3,1),
H (-1,2) and J (1,-3). Determine the coordinates of the vertices
of the triangle.

Help, Dr. math!
```

```
Date: 09/05/2002 at 11:31:45
From: Doctor Greenie
Subject: Re: Using midpoints to determine vertices

Hi, Leanne -

Let the vertices of the triangle be A, B, and C, such that given point
G(3,1) is the midpoint of AB, given point H(-1,2) is the midpoint of
AC, and given point J(1,-3) is the midpoint of BC. Let the points A,
B, and C be defined by

A(x1, y1)
B(x2, y2)
C(x3, y3)

Now, using the known coordinates of the midpoints G, H, and J, the
midpoint formula gives us the following equations relating the x- and
y-coordinates of points A, B, and C:

(x1+x2)/2 = 3;  (y1+y2)/2 = 1   [G(3,1) is the midpoint of AB]
(x1+x3)/2 = -1; (y1+y3)/2 = 2   [H(-1,2) is the midpoint of AC]
(x2+x3)/2 = 1;  (y2+y3)/2 = -3  [J(1,-3) is the midpoint of BC]

The three equations in x1, x2, and x3 give us a system of equations
that we can solve to find the values of x1, x2, and x3; and likewise
the three equations in y1, y2, and y3 give us a system of equations
that we can solve to find the values of y1, y2, and y3.

I hope this helps. Please write back if you have any further questions

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

```
Date: 09/17/2002 at 06:27:08
From: L. Hummel
Subject: Vectors of a triangle where the median points are known.

Hi,

I have a problem. I am searching for the three corners A, B, C, in a
triangle, where the midpoints of the triangle's sides are given:

M(AB) = (6,2)
M(AC) = (5,4)
M(BC) = (4,1)

How do I do this?

Best regards,
L. Hummel
```

```
Date: 09/17/2002 at 13:28:52
From: Doctor Greenie
Subject: Re: Vectors of a triangle where the median points are known.

Hello -

Let's choose some variables to denote the coordinates of the vertices
A, B, and C that we are looking for:

A(x1,y1)
B(x2,y2)
C(x3,y3)

Using the midpoint formula and the given coordinates of the midpoints
of sides AB, AC, and BC, we get a series of equations

(x1+x2)/2=6  (y1+y2)/2=2
(x1+x3)/2=5  (y1+y3)/2=4
(x2+x3)/2=4  (y2+y3)/2=1

This gives us a series of three equations in x1, x2, and x3 that we
can solve for those values, and it gives us a series of three
equations in y1, y2, and y3 that we can solve for those values.

The form of these equations gives us a chance to use a "trick" to
solve the sets of equations quite easily, as described below.

First, rewrite the equations without the "/2" in each one:

x1+x2=12  y1+y2=4
x1+x3=10  y1+y3=8
x2+x3= 8  y2+y3=2

Now see what happens when we add each set of three equations: We get

2x1+2x2+2x3=30   2y1+2y2+2y3=14
x1+x2+x3=15      y1+y2+y3=7

Now we can easily compare these last two equations with the sets of
equations to find immediately that

x3=3  x2=5  x1=7;  and  y3=3  y2=-1  y1=5

So the vertices of the triangle are

A(x1,y1) = A(7,5)
B(x2,y2) = B(5,-1)
C(x3,y3) = C(3,3)

Checking these results quickly using the midpoint formula, we have

M(AB) = (6,2)
M(AC) = (5,4)
M(BC) = (4,1)

which agrees with the given information.

I hope this helps.  Please write back if you have any further

- Doctor Greenie, The Math Forum

```
Associated Topics:
High School Coordinate Plane Geometry
High School Triangles and Other Polygons

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