Question

6 cm
? ______
6 cm
4 cm
? ______
10 cm
12 cm
57
shape 1 area __________
shape 2 area __________
shape 3 area __________
= ______cm²

Explanation:

Step1: Determine the base of the triangle (Shape 1)

The base of the triangle is equal to the length of the larger rectangle, which is \(12 - 4 = 8\) cm? Wait, no. Wait, the larger rectangle's width: Wait, the total length is 12 cm, and the small rectangle is 4 cm wide, so the base of the triangle (and the top of the larger rectangle) is \(12 - 4 = 8\) cm? Wait, no, looking at the diagram, the larger rectangle's width (the base of the triangle) is \(12 - 4 = 8\) cm? Wait, no, maybe I misread. Wait, the small rectangle is 4 cm wide, so the larger rectangle (the middle one) has a width of \(12 - 4 = 8\) cm. So the base of the triangle (Shape 1) is 8 cm. The height of the triangle is 6 cm. So area of triangle (Shape 1) is \(\frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 6 = 24\) \(cm^2\).

Step2: Area of the larger rectangle (Shape 2)

The larger rectangle has a height of 10 cm and a width of 8 cm (since total length is 12, small rectangle is 4, so 12 - 4 = 8). Wait, no, wait the small rectangle: the small rectangle's height: the larger rectangle's height is 10 cm, and the small rectangle's height is \(10 - 6 = 4\) cm? Wait, no, the small rectangle is 4 cm wide, and its height: looking at the diagram, the small rectangle is adjacent to the larger rectangle which is 10 cm tall, and the small rectangle's height is equal to \(10 - 6 = 4\) cm? Wait, maybe I made a mistake. Wait, the small rectangle: width 4 cm, height: let's see, the larger rectangle is 10 cm tall, and the small rectangle's height is \(10 - 6 = 4\) cm? Wait, no, the small rectangle's height is 4 cm? Wait, the diagram shows the small rectangle with 4 cm width and height? Wait, no, the larger rectangle is 10 cm tall, and the small rectangle is 4 cm wide, and its height is \(10 - 6 = 4\) cm? Wait, maybe the small rectangle's height is 4 cm? Wait, let's re-examine. The larger rectangle: height 10 cm, width 8 cm (12 - 4). The small rectangle: width 4 cm, height: the height of the small rectangle is equal to the height of the larger rectangle minus 6? No, the small rectangle is 4 cm wide, and its height is 4 cm? Wait, the diagram has a "?" for the small rectangle's height. Wait, the larger rectangle is 10 cm tall, and the small rectangle is 4 cm wide, and its height is \(10 - 6 = 4\) cm? Wait, maybe the small rectangle's height is 4 cm. So area of small rectangle (Shape 3) is \(4 \times 4 = 16\) \(cm^2\). Wait, no, maybe the larger rectangle (Shape 2) is 10 cm tall and 8 cm wide (12 - 4 = 8), so area is \(8 \times 10 = 80\) \(cm^2\)? Wait, no, that can't be, because then total area would be too big. Wait, maybe I messed up the shapes. Let's identify the shapes:

Shape 1: Triangle (top)
Shape 2: Larger rectangle (middle)
Shape 3: Small rectangle (left)

Wait, the small rectangle: width 4 cm, height: the height of the small rectangle is equal to the height of the larger rectangle minus 6? No, the larger rectangle is 10 cm tall, and the small rectangle is 4 cm wide, and its height is \(10 - 6 = 4\) cm? Wait, the diagram shows the small rectangle with 4 cm width and height? Wait, the "?" for the small rectangle's height: let's see, the larger rectangle is 10 cm tall, and the small rectangle is adjacent to it, with 6 cm marked on the larger rectangle. Wait, maybe the small rectangle's height is 4 cm (10 - 6 = 4). So small rectangle (Shape 3) has width 4 cm, height 4 cm, area \(4 \times 4 = 16\) \(cm^2\).

Larger rectangle (Shape 2): width 8 cm (12 - 4), height 10 cm, area \(8 \times 10 = 80\) \(cm^2\)? Wait, no, that can't be, because then total area would be 24 + 80 + 16 = 120, but maybe I made a mistake. Wait, no, the triangle's base: maybe the base of the triangle is 12 - 4 = 8 cm? Wait, the total length is 12 cm, the small rectangle is 4 cm wide, so the larger rectangle (and the triangle's base) is 12 - 4 = 8 cm. So triangle (Shape 1): base 8 cm, height 6 cm, area \(\frac{1}{2} \times 8 \times 6 = 24\) \(cm^2\).

Larger rectangle (Shape 2): width 8 cm, height 10 cm? Wait, no, the larger rectangle's height is 10 cm, but the small rectangle's height is 4 cm (since 10 - 6 = 4). Wait, maybe the larger rectangle's height is 10 cm, and the small rectangle's height is 4 cm (10 - 6 = 4). So larger rectangle (Shape 2): width 8 cm, height 10 cm? No, that would make the larger rectangle's area 8*10=80, small rectangle 4*4=16, triangle 24, total 24+80+16=120. But maybe I messed up the larger rectangle's height. Wait, the larger rectangle's height is 10 cm, and the small rectangle's height is 4 cm (10 - 6 = 4). So small rectangle (Shape 3): 4 cm (width) * 4 cm (height) = 16 \(cm^2\).

Wait, maybe the larger rectangle (Shape 2) is 8 cm (width) * 10 cm (height) = 80 \(cm^2\). Triangle (Shape 1): 24 \(cm^2\). Small rectangle (Shape 3): 4*4=16 \(cm^2\). Total area: 24 + 80 + 16 = 120 \(cm^2\).

Wait, let's recheck:

- Shape 1 (Triangle): base = 12 - 4 = 8 cm, height = 6 cm. Area = 0.5 * 8 * 6 = 24.

- Shape 2 (Larger Rectangle): width = 8 cm (12 - 4), height = 10 cm. Area = 8 * 10 = 80.

- Shape 3 (Small Rectangle): width = 4 cm, height = 10 - 6 = 4 cm. Area = 4 * 4 = 16.

Total area: 24 + 80 + 16 = 120.

Answer:

Shape 1 Area: \(24\) \(cm^2\)
Shape 2 Area: \(80\) \(cm^2\)
Shape 3 Area: \(16\) \(cm^2\)
Total Area: \(120\) \(cm^2\)