Rectangle Area Property

When a large rectangle is divided into four smaller rectangles by one horizontal and one vertical line, the areas of the smaller rectangles have a specific relationship. Let the areas be:

• $A_1$ (top-left)
• $A_2$ (top-right)
• $A_3$ (bottom-left)
• $A_4$ (bottom-right)

The property states that the product of the areas of diagonally opposite rectangles is equal: $A_1 \times A_4 = A_2 \times A_3$

Assigning Given Areas

The problem provides three areas from the diagram, indicated as '8124'. Based on the structure required to match the options, we identify the three given areas as 4, 8, and 12. We assign these values to find the unknown area:

• Top-left area, $A_1 = 4$
• Top-right area, $A_2 = 8$
• Bottom-left area, $A_3 = 12$

Calculating Fourth Area

Using the area property, we find the area of the bottom-right rectangle ($A_4$): $A_1 \times A_4 = A_2 \times A_3$ $4 \times A_4 = 8 \times 12$ $4 \times A_4 = 96$ $A_4 = \frac{96}{4}$ $A_4 = 24$

Summing Areas for Full Rectangle

The area of the full rectangle is the sum of the four smaller rectangle areas:

Total Area = $A_1 + A_2 + A_3 + A_4$

Total Area = $4 + 8 + 12 + 24$

Total Area = $48$

The area of the full rectangle is 48.