<p>The diagram shows a triangle with vertices \(A_1(x_1, y_1)\), \(A_2(x_2, y_2)\) and \(A_3(x_3, y_3)\). Verify the <strong>shoelace formula</strong> by showing that</p>
<p>Area of triangle \(A_1 A_2 A_3 = \frac{1}{2} \begin{vmatrix} x_1 & x_2 & x_3 & x_1 \\ y_1 & y_2 & y_3 & y_1 \end{vmatrix}\)</p>
<p style="margin-left: 8rem">\( \, = \frac{1}{2} |x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3|.\)</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1772356661462-ipqgeann44j.png" style="max-width: min(250px, 100%); height: auto; display: block;" alt="Triangle in a Cartesian plane" class="alignright custom-width" /></figure></div>

The diagram shows a triangle with vertices \(A_1(x_1, y_1)\), \(A_2(x_2, y_2)\) and \(A_3(x_3, y_3)\). Verify the **shoelace formula** by showing that

Area of triangle \(A_1 A_2 A_3 = \frac{1}{2} \begin{vmatrix} x_1 & x_2 & x_3 & x_1 \\ y_1 & y_2 & y_3 & y_1 \end{vmatrix}\)

\( \, = \frac{1}{2} |x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3|.\)

Coordinate Geometry

Question 537de9bb