Question - Calculating Area of a Triangle using Shoelace Formula
Solution:
To find the area of the triangle with vertices $$(-1, 4)$$, $$(3, 2)$$, and $$(5, 3)$$, you can use the Shoelace formula (also called the Gauss area formula). This formula can be applied when you have the coordinates of the triangle's vertices.The Shoelace formula states that the area of the triangle $$ A $$ can be found by:$$A = \frac{1}{2} \left| (x_1y_2 + x_2y_3 + x_3y_1) - (y_1x_2 + y_2x_3 + y_3x_1) \right|$$Where $$ (x_1, y_1), (x_2, y_2), $$ and $$ (x_3, y_3) $$ are the coordinates of the vertices.Let's plug in the coordinates:$$x_1 = -1, y_1 = 4 \\x_2 = 3, y_2 = 2 \\x_3 = 5, y_3 = 3$$$$A = \frac{1}{2} | ((-1) \cdot 2 + 3 \cdot 3 + 5 \cdot 4) - (4 \cdot 3 + 2 \cdot 5 + 3 \cdot (-1)) |$$Now multiply and add the corresponding terms:$$A = \frac{1}{2} | ((-2) + 9 + 20) - (12 + 10 - 3) |$$Simplify the terms inside the absolute value:$$A = \frac{1}{2} | 27 - 19 |$$$$A = \frac{1}{2} | 8 |$$$$A = \frac{1}{2} \cdot 8$$$$A = 4$$So the area of the triangle is 4 square units.
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