Calculate Area of a Quadrilateral Using the Shoelace Theorem

Math Problem Statement

Calculate the area of the quadrilateral ABCD using the given coordinates in the graph.

Solution

To calculate the area of the quadrilateral $ABCD$ in the coordinate plane, we can use the Shoelace Theorem. Here's the step-by-step approach:

Step 1: Identify the Coordinates

From the graph:

$A(3, -6)$
$B(2, 4)$
$C(-5, 3)$
$D(-4, -5)$

Step 2: Apply the Shoelace Formula

The formula for the area of a polygon with vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ is: $\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right|$ Here, $(x_{n+1}, y_{n+1}) = (x_1, y_1)$ to close the polygon.

Step 3: Compute the Determinant

Write the coordinates cyclically to close the polygon:

x & y \\ \hline 3 & -6 \\ 2 & 4 \\ -5 & 3 \\ -4 & -5 \\ 3 & -6 \\ \end{array}$$ Now compute: $$\text{Sum of products from top-left to bottom-right: } (3)(4) + (2)(3) + (-5)(-5) + (-4)(-6) = 12 + 6 + 25 + 24 = 67$$ $$\text{Sum of products from top-right to bottom-left: } (-6)(2) + (4)(-5) + (3)(-4) + (-5)(3) = -12 - 20 - 12 - 15 = -59$$ --- ### Step 4: Compute the Area $$\text{Area} = \frac{1}{2} \left| 67 - (-59) \right| = \frac{1}{2} \times 126 = 63 \, \text{square units}.$$ --- ### Final Answer: $$\boxed{63 \, \text{units}^2}$$ --- Would you like further clarification or a breakdown of any specific step? Let me know! Here are some follow-up questions for practice: 1. How is the Shoelace Theorem derived? 2. Can the area calculation change if we traverse the vertices in a different order? 3. How would you calculate the perimeter of the quadrilateral? 4. What happens if one vertex moves to another position? How does it affect the area? 5. Is the Shoelace Theorem applicable for a triangle as well? **Tip:** Always double-check the coordinates from the graph to avoid errors in calculation.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Polygon Area Calculation

Formulas

Shoelace Formula: Area = (1/2) * |Σ(x_i * y_(i+1) - y_i * x_(i+1))|

Theorems

Shoelace Theorem

Suitable Grade Level

Grades 9-11

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