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Shoelace Theorem

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The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.

Contents

1 Theorem
2 Proof
3 Problems
- 3.1 Introductory

Theorem

Suppose the polygon $P$ has vertices $(a_1, b_1)$ , $(a_2, b_2)$ , ... , $(a_n, b_n)$ , listed in clockwise order. Then the area of $P$ is

$\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|$

The Shoelace Theorem gets its name because if one lists the the coordinates in a column, $\begin{align*}(a_1 &, b_1) \\(a_2 &, b_2) \\& \vdots \\(a_n &, b_n) \\(a_1 &, b_1)\end{align*},$ and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.

Proof

This proof is incomplete. You can help us out by completing it.

Problems

Introductory

In right triangle $ABC$ , we have $\angle ACB=90^{\circ}$ , $AC=2$ , and $BC=3$ . Medians $AD$ and $BE$ are drawn to sides $BC$ and $AC$ , respectively. $AD$ and $BE$ intersect at point $F$ . Find the area of $\triangle ABF$ .

This article is a stub. Help us out by expanding it.

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Categories: Incomplete material | Stubs | Geometry | Theorems

Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

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