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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 
about any of this.

- 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?

Thank you in advance.

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 
questions about any of this.

- Doctor Greenie, The Math Forum
  
Associated Topics:
High School Coordinate Plane Geometry
High School Triangles and Other Polygons

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