To find the area of a quadrilateral given its vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$, we use the Shoelace Formula. Let's calculate the area step-by-step.

The vertices are:

• $A(1, 7)$
• $B(1, 11)$
• $C(8, 4)$
• $D(4, 4)$

Shoelace Formula:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$

Substitute the coordinates into the formula:

1. Calculate the products for the first sum ($x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1$): $1(11) + 1(4) + 8(4) + 4(7) = 11 + 4 + 32 + 28 = 75$

2. Calculate the products for the second sum ($y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1$): $7(1) + 11(8) + 4(4) + 4(1) = 7 + 88 + 16 + 4 = 115$

3. Compute the difference: $\left| 75 - 115 \right| = 40$

4. Divide by 2: $\text{Area} = \frac{1}{2} \times 40 = 20$

Final Answer:

The area of the quadrilateral is 20 square units.

Would you like a more detailed explanation or assistance with another question?

Related Questions:

1. How does the Shoelace Formula generalize to polygons with more than four vertices?
2. Why is the absolute value used in the formula?
3. How can you check if the quadrilateral is a rectangle or another specific type of quadrilateral?
4. Can this method be applied to three-dimensional shapes?
5. What are other methods for finding areas of irregular quadrilaterals?

Tip:

Always list vertices in a consistent order (clockwise or counterclockwise) to ensure the formula works correctly.