Question

question given the two rectangles below. find the area of the shaded region.

Explanation:

Step1: Find the length of the big rectangle

The big rectangle's length is the sum of the two 6s, so \( 6 + 6 = 12 \). The height of the big rectangle is \( 8 + 2 = 10 \)? Wait, no, wait. Wait, looking at the diagram, the white rectangle has height 8 and width 6, and the big rectangle's height is \( 8 + 2 = 10 \)? Wait, no, maybe I misread. Wait, the white rectangle is 8 (height) and 6 (width), and the big rectangle's length is \( 6 + 6 = 12 \), and the total height is \( 8 + 2 = 10 \)? Wait, no, maybe the big rectangle is formed by the white and the shaded. Alternatively, the area of the shaded region can be found by total area minus white area. Let's check:

Total area of the big rectangle: length is \( 6 + 6 = 12 \), height is \( 8 + 2 = 10 \)? Wait, no, the white rectangle is 8 (height) and 6 (width), and the bottom shaded part is 2 (height) and 12 (length), and the right shaded part is 6 (width) and 8 (height)? Wait, maybe better to calculate total area and subtract white area.

Total area: The big rectangle's length is \( 6 + 6 = 12 \), height is \( 8 + 2 = 10 \)? Wait, no, the white rectangle is 8 (height) and 6 (width), so the big rectangle's dimensions: length is \( 6 + 6 = 12 \), height is \( 8 + 2 = 10 \). So total area is \( 12 \times 10 = 120 \).

White area: The white rectangle is 8 (height) and 6 (width), so area is \( 8 \times 6 = 48 \).

Then shaded area is total area minus white area: \( 120 - 48 = 72 \). Wait, but let's check another way. The shaded region can be divided into two parts: the bottom rectangle (height 2, length 12) and the right rectangle (height 8, width 6). So bottom area: \( 12 \times 2 = 24 \), right area: \( 8 \times 6 = 48 \). Then total shaded area: \( 24 + 48 = 72 \). Yes, that matches.

So step by step:

Step1: Calculate total area of the big rectangle

Length of big rectangle: \( 6 + 6 = 12 \)
Height of big rectangle: \( 8 + 2 = 10 \)
Total area: \( 12 \times 10 = 120 \)

Step2: Calculate area of the white rectangle

White rectangle: height 8, width 6
Area: \( 8 \times 6 = 48 \)

Step3: Subtract white area from total area to get shaded area

Shaded area: \( 120 - 48 = 72 \)

Alternatively, divide shaded into two parts:

Bottom shaded: length 12, height 2. Area: \( 12 \times 2 = 24 \)
Right shaded: length 6, height 8. Area: \( 6 \times 8 = 48 \)
Total shaded: \( 24 + 48 = 72 \)

Answer:

\( \boxed{72} \)