Straight Lines
Area of Polygon using Coordinates
Grade 11
Question:
<p>The area of the pentagon whose vertices are <p>A(1, 1), B(7, 21), C(7, -3), D(12, 2) and E(0, -3), is</p>
<p>(a) \(\frac{137}{2}\) sq units</p>
<p>(b) \(\frac{137}{3}\) sq units</p>
<p>(c) \(\frac{137}{4}\) sq units</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the determinant formula for the area of a polygon by arranging vertices in order and applying the shoelace formula.
<p><strong>Solution:</strong></p><p>Using the formula for area of polygon with vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$:</p><p>$$\text{Area} = \frac{1}{2}\left|\begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \\ x_3 & y_3 \\ x_4 & y_4 \\ x_5 & y_5 \\ x_1 & y_1 \end{vmatrix}\right|$$</p><p>$$= \frac{1}{2}\left|\begin{vmatrix} 1 & 1 \\ 7 & 21 \\ 7 & -3 \\ 12 & 2 \\ 0 & -3 \\ 1 & 1 \end{vmatrix}\right|$$</p><p>$$= \frac{1}{2}|[(21 - 21 + 14 - 36 + 0) - (7 + 147 - 36 + 0 - 3)]|$$</p><p>$$= \frac{137}{2} \text{ sq units}$$</p><p>∴ Answer is (a)</p>
Correct Answer: A