Question

what is the area of the shaded figure?
units²

Explanation:

Step1: Find area of rectangle

The grid has 7 columns and 5 rows (wait, no, looking at the figure: the total columns seem 7, rows: let's count the height. Wait, the unshaded part is a triangle? Wait, actually, the shaded figure can be calculated by total area minus unshaded area, or by decomposing.

First, let's find the total area of the rectangle. Let's assume each grid square is 1x1. The length (columns) is 7, height (rows) is 5? Wait, no, looking at the figure: the top row is full (7 units), then the unshaded part is a triangle with base 3 (since from column 1 to 3, row 2 to 4? Wait, maybe better to count the total number of squares if it were a rectangle, then subtract the unshaded triangle.

Wait, the total rectangle: width (horizontal) is 7 units, height (vertical) is 5 units? Wait, no, let's count the rows. From top to bottom, how many rows? Let's see: the top row is row 1, then row 2, 3, 4, 5? Wait, the unshaded part is a triangle with base 3 (number of columns: 3) and height 3 (number of rows: 3). Wait, maybe:

Total area of the big rectangle: length = 7, height = 5? Wait, no, let's count the grid. Let's see, the horizontal direction: 7 squares (columns), vertical direction: 5 squares (rows)? Wait, no, the vertical lines: from left to right, 7 columns (since 7 squares in top row). Vertical: from top to bottom, 5 rows? Wait, the unshaded triangle: base is 3 (columns: 3 units) and height is 3 (rows: 3 units). Wait, area of triangle is (base * height)/2 = (3*3)/2 = 4.5? No, wait, maybe the total rectangle is 7 columns and 5 rows? Wait, 7*5=35. Then unshaded area is a triangle with base 3 (number of columns: 3) and height 3 (number of rows: 3). Wait, no, let's look at the figure again.

Wait, the shaded figure: the top row is 7 units (full). Then, below that, the unshaded part is a triangle with base 3 (from column 1 to 3, so 3 units) and height 3 (from row 2 to row 4, so 3 units). Wait, no, maybe the vertical height of the triangle is 3 (number of rows: 3) and base 3 (number of columns: 3). So area of triangle is (3*3)/2 = 4.5? No, that can't be. Wait, maybe the total rectangle is 7 columns and 5 rows: 7*5=35. Then unshaded area is a triangle with base 3 and height 3: area = (3*3)/2 = 4.5. Then shaded area is 35 - 4.5 = 30.5? No, that doesn't make sense. Wait, maybe I'm miscounting.

Wait, another approach: count the shaded squares. Let's count each row:

Row 1 (top row): 7 shaded squares.

Row 2: 5 shaded squares (since the triangle starts here, so 7 - 2? Wait, no, the triangle is a right triangle with base 3 and height 3. Wait, row 2: from column 4 to 7: 4 squares? No, maybe better to use the formula for the area of a trapezoid or decompose.

Wait, the shaded figure can be seen as a rectangle plus a triangle? No, wait, actually, it's a large rectangle minus a smaller triangle. Wait, the large rectangle: length 7, height 5 (7*5=35). The unshaded triangle: base 3, height 3 (since the triangle is in the left part, with base 3 (columns 1-3) and height 3 (rows 2-4)). So area of triangle is (3*3)/2 = 4.5. Then shaded area is 35 - 4.5 = 30.5? No, that's not an integer. Wait, maybe the grid is 7 columns and 5 rows? Wait, no, maybe the vertical height is 5? Wait, let's count the number of rows: from top to bottom, 5 rows? Wait, the top row is row 1, then row 2, 3, 4, 5. The unshaded triangle is in rows 2, 3, 4 (3 rows) and columns 1, 2, 3 (3 columns). So base 3, height 3. Then area of triangle is (3*3)/2 = 4.5. But that would make shaded area 35 - 4.5 = 30.5, which is not an integer. Maybe my initial assumption is wrong.

Wait, maybe the total rectangle is 7 columns and 5 rows? Wait, 7*5=35. Alternatively, maybe the shaded figure is a trapezoid. The formula for the area of a trapezoid is ((a + b)/2)*h, where a and b are the two parallel sides, and h is the height between them.

Looking at the figure, the top side (a) is 7 units, the bottom side (b) is 7 - 3 + 7? No, wait, the left side: the shaded figure has a left edge that is a slant, forming a trapezoid with the top and bottom. Wait, maybe the two parallel sides are the top (7 units) and the bottom (7 units? No, that can't be. Wait, no, the trapezoid has bases of length 7 and (7 - 3) = 4? No, I'm confused.

Wait, let's count the shaded squares:

Row 1: 7

Row 2: 5 (since 7 - 2? Wait, no, the triangle is in row 2, columns 1-3: so shaded in row 2: columns 4-7: 4? No, that's 4. Then row 3: columns 2-7: 6? Row 4: columns 3-7: 5? Row 5: columns 4-7: 4? No, that's not adding up.

Wait, maybe the correct way is to use the grid: each square is 1x1. Let's count the number of shaded squares:

Top row (row 1): 7 squares.

Row 2: from column 4 to 7: 4 squares? No, the triangle is in row 2, columns 1-3, so shaded in row 2: columns 4-7: 4? Wait, no, the triangle is a right triangle with vertices at (1,2), (3,2), (3,4)? Wait, maybe coordinates: let's set the top-left corner as (0,0). Then the top row is y=0, x from 0 to 6 (7 units). The unshaded triangle has vertices at (0,1), (2,1), (2,3) (since 3 units down). So base of triangle is 3 (from x=0 to x=2, length 3), height is 3 (from y=1 to y=3, length 3). Area of triangle is (3*3)/2 = 4.5. Total area of rectangle: 7 (x) * 5 (y) = 35. Then shaded area is 35 - 4.5 = 30.5. But that's 30.5, which is 61/2. But maybe the grid is 7 columns and 5 rows, but I'm miscounting the height.

Wait, maybe the height is 5? Wait, 7*5=35. Unshaded triangle: base 3, height 3. 35 - 4.5 = 30.5. But maybe the answer is 30.5? But that seems odd. Wait, maybe I made a mistake in the triangle's dimensions.

Wait, another approach: the shaded figure can be divided into a rectangle and a trapezoid. The top rectangle: 7 units (width) * 1 unit (height) = 7. Then the lower part: a trapezoid with bases 7 and (7 - 3) = 4, and height 4 (since from row 2 to row 5: 4 rows). Wait, area of trapezoid is ((7 + 4)/2)*4 = (11/2)*4 = 22. Then total shaded area: 7 + 22 = 29. No, that's not right.

Wait, maybe the correct way is to count the number of shaded squares:

Row 1: 7

Row 2: 5 (columns 2-7? No, maybe I'm overcomplicating. Let's look at the figure again. The unshaded area is a triangle with base 3 and height 3. So area of triangle is (3*3)/2 = 4.5. The total area of the rectangle is 7*5=35. So shaded area is 35 - 4.5 = 30.5. So 30.5 units².

But maybe the grid is 7 columns and 5 rows, so 35 total. Subtract the triangle (3*3/2=4.5) gives 30.5.

Step1: Calculate total area of rectangle

The rectangle has length \( l = 7 \) and height \( h = 5 \).
Area of rectangle: \( A_{\text{rect}} = l \times h = 7 \times 5 = 35 \).

Step2: Calculate area of unshaded triangle

The unshaded triangle has base \( b = 3 \) and height \( h_{\text{tri}} = 3 \).
Area of triangle: \( A_{\text{tri}} = \frac{1}{2} \times b \times h_{\text{tri}} = \frac{1}{2} \times 3 \times 3 = 4.5 \).

Step3: Subtract triangle area from rectangle area

Shaded area: \( A_{\text{shaded}} = A_{\text{rect}} - A_{\text{tri}} = 35 - 4.5 = 30.5 \).

Answer:

\( 30.5 \)