Difference between revisions of "Shoelace Theorem"
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=== Introductory === | === Introductory === | ||
In right triangle <math>ABC</math>, we have <math>\angle ACB=90^{\circ}</math>, <math>AC=2</math>, and <math>BC=3</math>. [[Median]]s <math>AD</math> and <math>BE</math> are drawn to sides <math>BC</math> and <math>AC</math>, respectively. <math>AD</math> and <math>BE</math> intersect at point <math>F</math>. Find the area of <math>\triangle ABF</math>. | In right triangle <math>ABC</math>, we have <math>\angle ACB=90^{\circ}</math>, <math>AC=2</math>, and <math>BC=3</math>. [[Median]]s <math>AD</math> and <math>BE</math> are drawn to sides <math>BC</math> and <math>AC</math>, respectively. <math>AD</math> and <math>BE</math> intersect at point <math>F</math>. Find the area of <math>\triangle ABF</math>. | ||
+ | |||
+ | == External Links== | ||
+ | A good explanation and exploration into why the theorem works: | ||
+ | [http://www.jamestanton.com/wp-content/uploads/2012/03/Cool-Math-Essay_June-2014_SHOELACE-FORMULA.pdf] | ||
Revision as of 14:34, 15 August 2015
The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.
Theorem
Suppose the polygon has vertices
,
, ... ,
, listed in clockwise order. Then the area of
is
The Shoelace Theorem gets its name because if one lists the coordinates in a column,
and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
Proof
Let be the set of points belonging to the polygon.
We have that
where
.
The volume form
is an exact form since
, where
Using this substitution, we have
Next, we use the theorem of Green to obtain
We can write
, where
is the line
segment from
to
. With this notation,
we may write
If we substitute for
, we obtain
If we parameterize, we get
Performing the integration, we get
More algebra yields the result
Problems
Introductory
In right triangle , we have
,
, and
. Medians
and
are drawn to sides
and
, respectively.
and
intersect at point
. Find the area of
.
External Links
A good explanation and exploration into why the theorem works: [1]