Length of the Diagonals of a Parallelogram

Date: 05/22/2000 at 19:46:31
From: moose
Subject: Calculus

A parallelogram has a 70-degree angle and sides 6cm and 10cm long. How 
long are its diagonals?

Date: 05/23/2000 at 11:53:07
From: Doctor Douglas
Subject: Re: Calculus

Hi Moose,

This is more of a trigonometry/geometry problem than a calculus 
problem. First we draw a parallelogram:

               10        
       A --------------- P(x,y)
        /              /
     6 /              / 6
      /_70deg________/
             10      B

Now, what is the length of say, the long diagonal? If we superimpose 
a Cartesian xy grid on this with the origin at the lower left, we then 
need to find the coordinates of the point P(x,y) at the upper right. 
We use trigonometry (see lower left angle, left side) to compute these 
coordinates:

    x = (bottom of triangle at left) + (upper edge)
      = (6 cos(70 deg)) + (10)

    y = (height of triangle at left)
      = (6 sin(70 deg))

So the long diagonal is sqrt(x^2 + y^2), by the Pythagorean theorem. 
See if you can use similar reasoning to obtain the coordinates of 
points A and B to find the length of the short diagonal.

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