Find the Fourth Side

Date: 12/03/2001 at 20:48:36
From: Suzy
Subject: Quadrilaterals

The successive sides of a quadrilateral are 2,6,9, and x. If the 
diagonals of the quadrilateral are perpendicular, compute x.

Date: 12/04/2001 at 08:17:15
From: Doctor Floor
Subject: Re: quadrilaterals

Hi, Suzy,

Thanks for your question.

The diagonals of the quadrilateral divide the figure into four right 
triangles:

  

From the three right triangles with known hypotenuses we see that:

  t^2 + u^2 = 4    [1]
  u^2 + v^2 = 36   [2]
  v^2 + w^2 = 81   [3]

Now, if we compute [1]-[2]+[3] we get:

  t^2 + u^2 - u^2 - v^2 + v^2 + w^2 = 4 - 36 + 81
  t^2 + w^2 = 49

And thus we see that x must be equal to 7.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/