Shoelace Theorem
Lesson · Intermediate
Geometry: Coordinate Geometry
The vertices should be written in clockwise or counterclockwise order:
The area of the polygon is:
The triangle with vertices
has area 20. Find .
Shoelace gives
Thus
Therefore,
Hence
Show that
are collinear.
Using shoelace,
Thus
Therefore,
A triangle has vertices
Point lies on
and satisfies
Find all possible areas of triangle .
Let
Since ,
So
Using the quadratic formula,
Therefore,
gives
So there are two possible positions for .
Now use Shoelace:
Thus
For
we get
For
we get
Therefore, the two possible areas are
The vertices must be written consecutively in either clockwise or counterclockwise order.
For example, using
when the actual boundary order is
can give the wrong result.
Do not simply sort points by -coordinate or -coordinate.
For hand calculations, rewrite the first coordinate pair at the bottom
Then multiply diagonally.
The polygon does not need to be convex