Is This a Square?

Date: 01/30/2001 at 12:48:25
From: Greg
Subject: Given points on graph, determine if it's a square

Given four points on a graph, what can I do to verify this is a 
square?

The distance of a line is:  sqrt[(x1 - x2)^2 + (y1 - y2)^2]

Do I pick one (x,y) reference point and calculate the distance to each 
of the three other points, verifying they are all equal in length?

Greg

Date: 01/30/2001 at 16:34:53
From: Doctor Rick
Subject: Re: Given points on graph, determine if it's a square

Hi, Greg.

No. If you did so, you'd be comparing two sides of the quadrilateral 
and one diagonal.

You need to find the length of each side (the distance between two 
adjacent vertices) and show that all four are equal. You also must 
show that each side is perpendicular to the two adjacent sides. This 
is true if the product of the slopes of the two lines is -1.

You don't have to do every one of these calculations. For instance, 
once you've shown that the four sides are of equal length, it's enough 
to show that a single pair of adjacent sides is perpendicular. A 
rhombus with one right angle is a square.

You can also find the center of one diagonal, (xc, yc), and show that 
the four vertices follow this pattern, with some constants a and b:

  (xc+a, yc+b); (xc+b, yc-a); (xc-a, yc-b); (xc-b, yc+a)

The center (xc, yc) is found by

  xc = (x3 - x1)/2
  yc = (y3 - y1)/2

where (x1, y1) and (x3, y3) are either pair of opposite vertices of 
the quadrilateral.

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

Date: 01/30/2001 at 16:39:12
From: Doctor Greenie
Subject: Re: Given points on graph, determine if its' a square

Hi, Greg --

Here is a rough picture of what you are working with...

                        B
  W +-------------------*--------+ X
    |               *    *       |
    |           *         *      |
    |       *              *     |
    |   *                   *    |
  A *                        *   |
    |*                        *  |
    | *                        * |
    |  *                        *|
    |   *                        * C
    |    *                   *   |
    |     *              *       |
    |      *         *           |
    |       *    *               |
  Z +--------*-------------------+ Y
             D

You are given points A, B, C, and D and are asked to determine if they 
form a square.

The method you suggest won't work. If you pick point A as your 
reference point, then the distances to points B and D should be the 
same (the length of the side of the square), but the distance to point 
C is the length of a diagonal of the square, which is different from 
the length of the sides.

For purposes of my discussion, I have created the rectangle WXYZ from 
vertical segments WZ and XY, containing points A and C, respectively, 
and horizontal segments WX and YZ, containing points B and D, 
respectively. (Actually, if ABCD is a square, then WXYZ is not just 
a rectangle but a square....) The sides of this rectangle are thus 
parallel to the x- and y-axes (the actual location of the x- and 
y-axes is irrelevant to the discussion).

To show that ABCD is a square, you need to show that AB, BC, CD, and 
DA are all the same length; and you need to show that the angles at A, 
B, C, and D are all right angles.

You could use the distance formula to show that AB, BC, CD, and DA are 
all the same length.  And if you remember the idea of slope from 
algebra, you could show that each angle is a right angle by showing 
that the slopes of the two lines meeting at each angle are negative 
reciprocals of each other (for example, the slopes are -2 and +1/2).

But in fact you don't need any of those sophisticated mathematical 
methods. You don't need to know about slopes, and you don't need to 
know the distance formula.

All you need to do to show that ABCD is a square is to show that the 
lengths AW, BX, CY, and DZ are all equal and that the lengths WB, XC, 
YD, and ZA are all equal. If these conditions hold, then the distance 
formula will show the lengths AB, BC, CD, and DA to be equal; and the 
slope formula will show that the line segments at each corner meet at 
right angles - so you will be done.

It is easy to measure the lengths of all these segments, because they 
are all either vertical or horizontal.

So to show the figure is a square, you could, for example, measure the 
lengths AW and WB; then the figure is a square if and only if (1) BX, 
CY, and DZ are all the same length as AW and (2) XC, YD, and ZA are 
all the same length as WB.

I hope this helps.  Write back if my words have confused you.

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