Question

find the area of the figure below by first finding the areas of the rectangle and triangles.
(grid with 1 unit scale, figure composed of rectangle and triangles)
area of the rectangle:
blank square unit
area of the triangle on the top:
blank square unit
area of the triangle on the bottom:
blank square unit
area of the figure:
blank square unit

Explanation:

Step1: Determine rectangle dimensions

From the grid, the rectangle has length \( l = 4 \) units and height \( h = 3 \) units. Area of rectangle: \( A_{rect} = l \times h = 4 \times 3 = 12 \) square units.

Step2: Determine triangle dimensions (left and right)

Each triangle has base \( b = 3 \) units and height \( h = 3 \) units. Area of one triangle: \( A_{tri} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 3 \times 3 = 4.5 \) square units. There are two triangles, so total triangle area: \( 2 \times 4.5 = 9 \) square units. Wait, no—wait, looking at the figure, actually, the left and right triangles: wait, maybe I misread. Wait, the figure is a trapezoid, composed of a rectangle and two triangles? Wait, no, maybe the top triangle? Wait, no, the problem says "first finding the areas of the rectangle and triangles". Wait, maybe the rectangle is in the middle, and two triangles (left and right). Wait, let's re-examine. The grid: each square is 1 unit. The rectangle: length 4, height 3 (so area 12). Then the left triangle: base 3, height 3? Wait, no, maybe the two triangles (left and right) each have base 3 and height 3? Wait, no, maybe the top triangle? Wait, no, the problem's table: "Area of the rectangle", "Area of the triangle on the top", "Area of the triangle on the bottom", "Area of the figure". Wait, maybe the figure is a trapezoid, with a rectangle and two triangles (top and bottom? No, maybe left and right). Wait, perhaps I made a mistake. Wait, let's count the grid. The rectangle: length 4 (from x=3 to x=7, 4 units) and height 3 (from y=2 to y=5, 3 units). Then the left triangle: base 3 (from x=0 to x=3, 3 units) and height 3 (same y-range). The right triangle: base 3 (from x=7 to x=10, 3 units) and height 3. Wait, but then area of rectangle is 4*3=12. Area of one triangle: (3*3)/2=4.5, two triangles: 9. Then total area: 12+9=21? But maybe the top triangle? No, the problem's table has "Area of the triangle on the top" and "Area of the triangle on the bottom". Wait, maybe I misinterpret. Wait, perhaps the figure is a trapezoid with bases: top base 4, bottom base 10 (4+3+3), height 3. Area of trapezoid: \( \frac{(a + b)}{2} \times h = \frac{(4 + 10)}{2} \times 3 = 21 \). But using rectangle and two triangles: rectangle (4x3=12), two triangles (each 3x3/2=4.5, so 9), total 21. But the problem's table: "Area of the rectangle: 12", "Area of the triangle on the top: wait, no, maybe the top triangle is different. Wait, no, the user's problem: let's re-express.

Wait, maybe the rectangle is 4x3=12. Then the two triangles (left and right) each have base 3 and height 3? Wait, no, maybe the left triangle: base 3, height 3, area 4.5; right triangle: base 3, height 3, area 4.5. Then total area: 12 + 4.5 + 4.5 = 21. But the problem's table has "Area of the triangle on the top" and "Area of the triangle on the bottom"—maybe that's a typo, and it's left and right. Alternatively, maybe the top triangle is zero? No. Wait, perhaps I made a mistake. Let's do it step by step as per the problem's instruction:

1. Area of the rectangle: length 4, height 3 (since each square is 1 unit). So \( 4 \times 3 = 12 \) square units.

2. Area of the triangle on the top: Wait, maybe the top triangle is not present. Wait, no, the figure: looking at the grid, the top is a horizontal line, then left and right slanting lines. Wait, maybe the figure is a trapezoid, with the rectangle in the middle, and two triangles (left and right) each with base 3 and height 3. So area of each triangle: \( \frac{1}{2} \times 3 \times 3 = 4.5 \). So two triangles: \( 4.5 \times 2 = 9 \). Then total area: \( 12 + 9 = 21 \). But the problem's table has "Area of the triangle on the top" and "Area of the triangle on the bottom"—maybe that's an error, and it's left and right. Alternatively, maybe the top triangle is a different one. Wait, maybe the user's figure: the rectangle is 4x3 (12), and two triangles (each with base 3 and height 3), so total area 21. But let's check again.

Wait, maybe the left triangle: base 3, height 3 (area 4.5), right triangle: base 3, height 3 (area 4.5). So total area: 12 + 4.5 + 4.5 = 21. So:

- Area of the rectangle: 12 square units.

- Area of one triangle (e.g., left): 4.5, another (right): 4.5. But the problem says "Area of the triangle on the top" and "Area of the triangle on the bottom"—maybe that's a mistake, and it's left and right. Alternatively, maybe the top triangle is zero, but no. Wait, perhaps I misread the grid. Let's count the squares. The rectangle: 4 columns (x from 3 to 7) and 3 rows (y from 2 to 5), so 4*3=12. The left triangle: from (0,5) to (3,2) to (0,2): base 3 (x from 0 to 3), height 3 (y from 2 to 5). Area: 0.5*3*3=4.5. The right triangle: from (7,5) to (10,2) to (7,2): base 3, height 3. Area: 4.5. So total area: 12 + 4.5 + 4.5 = 21. So:

- Area of the rectangle: 12

- Area of the triangle on the top: Wait, no, maybe the top triangle is not there. Wait, the problem's table has four boxes: Area of rectangle, Area of triangle on top, Area of triangle on bottom, Area of figure. Maybe the two triangles are top and bottom? No, that doesn't make sense. Alternatively, maybe the figure is a trapezoid with bases 4 and 10 (4+3+3) and height 3. Area of trapezoid: \( \frac{(4 + 10)}{2} \times 3 = 21 \), which matches. So using rectangle (12) and two triangles (each 4.5, total 9), sum to 21.

So:

- Area of the rectangle: \( 4 \times 3 = 12 \) square units.

- Area of one triangle (e.g., left): \( \frac{1}{2} \times 3 \times 3 = 4.5 \) square units.

- Area of the other triangle (right): same as left, 4.5 square units.

- Area of the figure: \( 12 + 4.5 + 4.5 = 21 \) square units.

But the problem's table labels: "Area of the triangle on the top" and "Area of the triangle on the bottom"—maybe that's a mistake, and it's left and right. Assuming that, then:

Answer:

• Area of the rectangle: 12 square units
• Area of the triangle on the top (left triangle): 4.5 square units
• Area of the triangle on the bottom (right triangle): 4.5 square units
• Area of the figure: 21 square units

(Note: If the triangles are considered top and bottom, maybe my initial analysis is wrong. But based on the grid, the left and right triangles make sense. Alternatively, maybe the top triangle is different. But given the problem's instruction to find area by rectangle and triangles, the above is the calculation.)