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Problem 678

# Find the area of the polygon whose vertices are $$\mathrm{A}(2,2)$$, $$\mathrm{B}(9,3), \mathrm{C}(7,6)$$, and $$\mathrm{D}(4,5)$$.

Expert verified
The area of the polygon ABCD with vertices A(2,2), B(9,3), C(7,6), and D(4,5) is 15 square units.
See the step by step solution

## Step 1: Calculate the Shoelace Formula

The Shoelace formula can be expressed as follows: $Area = \frac{1}{2}\left| \sum_{i=1}^{n} x_iy_{i+1} - \sum_{i=1}^{n} y_ix_{i+1} \right|$ where $$(x_i, y_i)$$ are the coordinates of the vertices, and $$(x_{n+1},y_{n+1}) = (x_1, y_1)$$. For the given polygon with vertices A (2,2), B (9,3), C (7,6), and D (4,5), the formula becomes: $Area = \frac{1}{2}\left| (2 \cdot 3) + (9 \cdot 6) + (7 \cdot 5) + (4 \cdot 2) - (2 \cdot 9) - (3 \cdot 7) - (6 \cdot 4) - (5 \cdot 2) \right|$

## Step 2: Simplify the Expression

Now, simplify the expression by doing the multiplications and additions within the absolute value brackets: $Area = \frac{1}{2}\left| 6 + 54 + 35 + 8 - 18 - 21 - 24 - 10 \right|$

## Step 3: Calculate the Area

Calculate the area by adding and subtracting the remaining values and then multiply by 1/2: $Area = \frac{1}{2}\left| 103 - 73 \right|$ $Area = \frac{1}{2}(30)$ Finally, calculate the area of the polygon: $Area = 15$ The area of the polygon ABCD is 15 square units.

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