Teach me this question---**Question 9**
**Question Stem:**
A diagram of a flower garden is shown below.
What is the total area, in square feet, of the flower garden?
**Chart/Diagram Description:**
* **Type:** Geometric figure (polygon), specifically an L-shaped figure representing a flower garden.
* **Main Elements:**
* The figure is an L-shaped polygon, which can be decomposed into two rectangles or considered a larger rectangle with a smaller rectangular section removed.
* **Labeled Dimensions:**
* The top horizontal length of the entire figure is 9 ft.
* The top-left vertical length is 3 ft.
* The vertical length of the inner 'cut-out' part is 4 ft.
* The bottom horizontal length of the main part (to the right of the inner 'cut-out') is 6 ft.
* All visible corners are right angles.
* Units for all dimensions are "ft" (feet).
**Options:**
A 22
B 27
C 51
D 54
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Let's solve this L-shaped flower garden area problem. We have an L-shaped figure with given dimensions: 9 feet across the top, 3 feet on the left vertical side, 4 feet for the inner vertical cut, and 6 feet for the bottom horizontal section. This is a composite shape that can be solved using two different approaches: the addition method where we break it into separate rectangles, or the subtraction method where we consider it as a large rectangle with a piece removed.
Let's solve this using the addition method by breaking the L-shape into two separate rectangles. Rectangle 1 is the top portion, measuring 9 feet by 3 feet, giving us an area of 27 square feet. For Rectangle 2, we need to find the missing height dimension. Since the total vertical length is 7 feet and the top rectangle uses 3 feet, the remaining height is 4 feet. So Rectangle 2 measures 6 feet by 4 feet, giving us 24 square feet. Adding both rectangles together: 27 plus 24 equals 51 square feet total.
Understanding how to find missing dimensions is crucial for solving composite shape problems. Let's analyze the dimensional relationships systematically. First, we can determine the total vertical length by adding the given vertical dimensions: 3 feet plus 4 feet equals 7 feet total height. Next, we find the missing horizontal dimension of the cutout by subtracting: 9 feet minus 6 feet equals 3 feet. Finally, we can calculate the height of the bottom rectangle: 7 feet total minus 3 feet for the top section equals 4 feet. With these calculations, we now have all the dimensions needed to solve the problem.
Now let's solve the same problem using the subtraction method. We consider the L-shape as a large rectangle with a rectangular piece removed. The large rectangle measures 9 feet by 7 feet, giving us 63 square feet. The cutout rectangle has a width of 3 feet, which we calculated by subtracting 6 feet from 9 feet, and a height of 4 feet. This cutout area is 3 times 4, which equals 12 square feet. Subtracting the cutout from the large rectangle: 63 minus 12 equals 51 square feet, confirming our previous answer.
Let's verify our solution by comparing both methods. Method 1 using addition gave us 27 plus 24 equals 51 square feet. Method 2 using subtraction gave us 63 minus 12 equals 51 square feet. Both methods confirm the same answer. Looking at the multiple choice options: A is 22, B is 27, C is 51, and D is 54. Our calculated answer of 51 square feet corresponds to option C. This demonstrates the importance of using systematic approaches and verifying results through alternative methods when solving composite shape area problems.