Difference between revisions of "Shoelace Theorem"
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==Theorem== | ==Theorem== | ||
| − | Suppose the polygon <math>P</math> has vertices <math>(a_1, b_1)</math>, <math>(a_2, b_2)</math>, ... , <math>(a_n, b_n)</math>, listed in clockwise order. Then area of <math>P</math> is | + | Suppose the polygon <math>P</math> has vertices <math>(a_1, b_1)</math>, <math>(a_2, b_2)</math>, ... , <math>(a_n, b_n)</math>, listed in clockwise order. Then the area of <math>P</math> is |
<cmath>\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|</cmath> | <cmath>\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|</cmath> | ||
Revision as of 22:28, 18 May 2009
The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.
Contents
Theorem
Suppose the polygon 
has vertices 
, 
, ... , 
, listed in clockwise order. Then the area of is
![\[\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|\]](http://latex.artofproblemsolving.com/9/f/5/9f5392e8f69d97540c2fad3437685d2a8c6c19cb.png)
The Shoelace Theorem gets its name because if one lists the the coordinates in a column,
and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
Proof
Problems
Introductory
In right triangle 
, we have 
, 
, and 
. Medians 
and 
are drawn to sides 
and 
, respectively. and intersect at point 
. Find the area of 
.
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