Find the Area of the Quadrilateral

Date: 05/20/2003 at 00:18:57
From: N. Arun
Subject: Geometry

Vertex Q of square QRST is at the center of square MNOP. The length 
of a side of MNOP is 1 and the length of a side of QRST is 2. Side QR 
intersects OP at X and QT intersects MP at Y. If angle QXP is 60, the 
area of quadrilateral QXPY is 

[1] ((3^0.5)/6)  [2] 1/3  [3] 1/4  [4] ((2^0.5)/4)

I don't know how to calculate the area of a quadrilateral using only 
the vertex angles.

Date: 05/21/2003 at 12:28:09
From: Doctor Peterson
Subject: Re: Geometry

Thanks for writing to Dr. Math.

Here is my version of the figure; since the size of QRST doesn't 
matter, I only show Q and the edges connected to it:

    N-------------------------------------------O
    |                                           |
    |                                           |
    |                                           |     ----
    |                                           | ----
    |                                          -X-
    |                                      ----.|
    |                                  ----.....|
    |                               ---.........|
    |                           ----............|
    |                       ----................|
    |                     Q---------------------A
    |                     |\....................|
    |                     | \...................|
    |                     |  \..................|
    |                     |   \.................|
    |                     |    \................|
    |                     |     \...............|
    |                     |      \..............|
    |                     |       \.............|
    |                     |        \............|
    |                     |         \...........|
    M---------------------B----------Y----------P
                                      \
                                       \

Think about how you can find the areas of triangles QAX and QBY; 
don't actually calculate them just yet. How would you use those areas 
to find the area of QXPY? One approach is to compare it with square 
QAPB.

Do you see a very easy way to find the area?

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