Question

find the area of the figure. (sides meet at right angles.)

Explanation:

Step1: Divide the figure into two rectangles

We can split the composite figure into a vertical rectangle (right part) and a horizontal rectangle (bottom part). The vertical rectangle has dimensions \(3\,\text{ft}\) (width) and \(6\,\text{ft}\) (height)? Wait, no, wait. Wait, the bottom part: the total length is \(7\,\text{ft}\), and the top part is \(3\,\text{ft}\), so the bottom rectangle's length is \(7\,\text{ft}\), and its height is \(6 - 4=2\,\text{ft}\)? Wait, no, let's re - examine.

Alternative approach: Split into two rectangles. One rectangle is \(3\,\text{ft}\times6\,\text{ft}\)? No, wait, the left - bottom rectangle: length is \(4\,\text{ft}\), and height is \(6 - 4 = 2\,\text{ft}\)? Wait, no, the figure has a right angle, so we can split it into a rectangle with length \(7\,\text{ft}\) and height \(2\,\text{ft}\) (the bottom part) and a rectangle with length \(3\,\text{ft}\) and height \(4\,\text{ft}\) (the top - right part).

Wait, let's calculate the dimensions correctly. The total height of the figure is \(6\,\text{ft}\). The height of the bottom rectangle: since the top rectangle has a height of \(4\,\text{ft}\), the bottom rectangle has a height of \(6 - 4=2\,\text{ft}\), and its length is \(7\,\text{ft}\). The top - right rectangle has a length of \(3\,\text{ft}\) and a height of \(4\,\text{ft}\).

Step2: Calculate the area of each rectangle

- Area of the bottom rectangle: The formula for the area of a rectangle is \(A = l\times w\) (length \(\times\) width or length \(\times\) height). For the bottom rectangle, \(l = 7\,\text{ft}\), \(w=2\,\text{ft}\), so \(A_1=7\times2 = 14\,\text{ft}^2\).
- Area of the top - right rectangle: \(l = 3\,\text{ft}\), \(w = 4\,\text{ft}\), so \(A_2=3\times4=12\,\text{ft}^2\).

Step3: Sum the areas of the two rectangles

The total area \(A = A_1+A_2\). Substitute the values of \(A_1\) and \(A_2\): \(A=14 + 12=30\,\text{ft}^2\).

Another way: We can also split it into a rectangle of \(4\,\text{ft}\times2\,\text{ft}\) (left - bottom) and a rectangle of \(3\,\text{ft}\times6\,\text{ft}\) (right - side). Let's check: \(4\times2=8\), \(3\times6 = 18\), \(8 + 18=26\)? No, that's wrong. Wait, my first split was wrong.

Wait, let's look at the horizontal lengths. The total length of the base is \(7\,\text{ft}\), and the top horizontal length is \(3\,\text{ft}\), so the length of the left - bottom rectangle is \(7 - 3=4\,\text{ft}\), and its height is \(6 - 4 = 2\,\text{ft}\)? No, the height of the left - bottom rectangle: the total height is \(6\,\text{ft}\), and the height of the top rectangle is \(4\,\text{ft}\), so the height of the bottom rectangle is \(6-4 = 2\,\text{ft}\), and its length is \(7\,\text{ft}\). The top rectangle has length \(3\,\text{ft}\) and height \(4\,\text{ft}\). Wait, \(7\times2=14\), \(3\times4 = 12\), \(14 + 12=26\)? No, that's not right. Wait, maybe I made a mistake in the height of the bottom rectangle.

Wait, let's use the other method. The figure can be considered as a large rectangle minus a smaller rectangle. The large rectangle would have dimensions \(7\,\text{ft}\times6\,\text{ft}\), but there is a missing part. The missing part has a length of \(4\,\text{ft}\) and a height of \(4\,\text{ft}\) (since \(7 - 3=4\) and the height of the missing part is \(4\,\text{ft}\)).

Area of large rectangle: \(A_{large}=7\times6 = 42\,\text{ft}^2\)

Area of missing rectangle: \(A_{missing}=4\times4 = 16\,\text{ft}^2\)

Total area \(A = A_{large}-A_{missing}=42 - 16 = 26\,\text{ft}^2\)? No, that's not matching. Wait, I think I messed up the missing part.

Wait, let's look at the figure again. The figure has a horizontal segment of \(4\,\text{ft}\) on the left, then a vertical segment of \(4\,\text{ft}\) up, then a horizontal segment of \(3\,\text{ft}\) to the right, and then down \(6\,\text{ft}\) and left \(7\,\text{ft}\). So the correct way is to split into two rectangles:

1. Rectangle 1: Width \(3\,\text{ft}\), height \(6\,\text{ft}\)? No, because the left part has a length of \(4\,\text{ft}\). Wait, no, the bottom rectangle: length is \(7\,\text{ft}\), height is \(6 - 4=2\,\text{ft}\) (since the vertical side on the left - top is \(4\,\text{ft}\), and the total vertical side is \(6\,\text{ft}\)). Then the top rectangle: length is \(3\,\text{ft}\), height is \(4\,\text{ft}\). Then area of bottom rectangle: \(7\times2 = 14\), area of top rectangle: \(3\times4 = 12\), total \(14 + 12=26\)? Wait, no, maybe the height of the bottom rectangle is \(6 - 4 = 2\), and the length of the bottom rectangle is \(7\), and the top rectangle has length \(3\) and height \(4\). But \(7\times2=14\), \(3\times4 = 12\), \(14 + 12 = 26\). Wait, but let's check with another split.

Split into a rectangle of \(4\,\text{ft}\times(6 - 4)\,\text{ft}\) (left - bottom) and a rectangle of \(3\,\text{ft}\times6\,\text{ft}\) (right - side). Left - bottom: \(4\times2 = 8\), right - side: \(3\times6=18\), \(8 + 18 = 26\). Yes, that's correct.

Wait, I think my first mistake was in the height of the bottom rectangle. The vertical distance from the bottom to the horizontal segment (the one with \(4\,\text{ft}\) height) is \(6 - 4=2\,\text{ft}\). So the left - bottom rectangle has length \(4\,\text{ft}\) and height \(2\,\text{ft}\), area \(4\times2 = 8\). The right - side rectangle has length \(3\,\text{ft}\) and height \(6\,\text{ft}\), area \(3\times6 = 18\). Total area \(8 + 18=26\,\text{ft}^2\).

Wait, let's do it step by step correctly:

Step1: Identify the two rectangles

- Rectangle 1: The left - bottom part. Its length is \(4\,\text{ft}\) (horizontal) and its height is \(6 - 4=2\,\text{ft}\) (vertical, since the vertical segment above it is \(4\,\text{ft}\) and the total vertical length is \(6\,\text{ft}\)).
- Rectangle 2: The right - side part. Its length is \(3\,\text{ft}\) (horizontal) and its height is \(6\,\text{ft}\) (vertical).

Step2: Calculate the area of each rectangle

- Area of Rectangle 1: \(A_1=\text{length}\times\text{height}=4\times2 = 8\,\text{ft}^2\)
- Area of Rectangle 2: \(A_2=\text{length}\times\text{height}=3\times6 = 18\,\text{ft}^2\)

Step3: Sum the areas

Total area \(A = A_1+A_2=8 + 18=26\,\text{ft}^2\)

Wait, now I am confused. Let's use the coordinate system approach. Let's place the bottom - left corner at \((0,0)\). Then:

- The bottom - right corner is at \((7,0)\).
- The top - right corner is at \((7,6)\).
- The left - top corner of the indent is at \((4,4)\) (since from \(x = 4\) to \(x = 7\) is \(3\,\text{ft}\), and from \(y = 4\) to \(y = 6\) is \(2\,\text{ft}\)? No, wait, the vertical segment from \((4,0)\) to \((4,4)\) (length \(4\,\text{ft}\)), then from \((4,4)\) to \((7,4)\) (length \(3\,\text{ft}\)), then from \((7,4)\) to \((7,6)\) (length \(2\,\text{ft}\)), then from \((7,6)\) to \((0,6)\)? No, no, the figure is: bottom side from \((0,0)\) to \((7,0)\), left side from \((0,0)\) to \((0,2)\) (wait, no, I think I messed up the coordinates).

Wait, the correct way: The figure has a right angle, so we can split it into two rectangles:

1. The lower rectangle: length \(7\,\text{ft}\), height \(h_1\). The upper rectangle: length \(3\,\text{ft}\), height \(h_2 = 4\,\text{ft}\). And \(h_1+h_2=6\,\text{ft}\), so \(h_1 = 6 - 4=2\,\text{ft}\). So area of lower rectangle: \(7\times2 = 14\), area of upper rectangle: \(3\times4 = 12\), total \(14 + 12=26\).

Yes, that's correct. So the area is \(26\) square feet.

Answer:

\(26\)