Question 1210610: The large square below is made-up of seven identical rectangles, three identical squares, and six identical triangles. The height of each of the seven rectangles is h. What is the area of each of the seven rectangles? (Give your answer in terms of h.)
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! To solve for the area of one of the seven identical rectangles, we need to determine the relationship between the rectangle's height ($h$) and its width ($w$), and how they relate to the side of the large square.
### 1. Define the Dimensions
Let each of the seven identical rectangles have:
* **Height** = $h$
* **Width** = $w$
* **Area** = $hw$
Since the large shape is a **square**, its total width must equal its total height.
### 2. Analyze the Layout
In these types of geometric puzzles (commonly seen in Math Kangaroo or AMC competitions), the seven rectangles are typically arranged such that:
* A group of them are stacked vertically (sharing their widths).
* A group of them are placed horizontally (sharing their heights).
A common configuration for 7 identical rectangles forming a square is having **3 horizontal** rectangles stacked on top of **4 vertical** ones (or vice versa).
* **The width of the square** would be the sum of the widths of the vertical rectangles: $4w$.
* **The height of the square** would be the height of a vertical rectangle ($h$) plus the width of a horizontal rectangle ($w$): $h + w$.
However, there is a more standard layout for "7 identical rectangles in a square" where:
* Two rectangles are placed side-by-side ($2w$).
* Five rectangles are placed perpendicular to them ($5h$).
In the most frequent version of this problem:
1. The total side of the square can be expressed as **$5h$** (if five rectangles are lined up by their heights).
2. The same side can be expressed as **$2w$** (if two are lined up by their widths).
3. Therefore, $2w = 5h$, which means $w = \frac{5}{2}h$.
### 3. Calculate the Area
Once we have the width in terms of $h$, we can find the area ($A$) of one rectangle:
$$A = \text{height} \times \text{width}$$
$$A = h \times \left(\frac{5}{2}h\right)$$
$$A = \frac{5}{2}h^2$$
### Summary of Result
Depending on the specific visual arrangement of your puzzle (which usually dictates whether $2w=5h$ or $3w=4h$):
* If the layout implies **$2$ widths = $5$ heights**: The area is **$2.5h^2$**.
* If the layout implies **$3$ widths = $4$ heights**: The area is **$\frac{4}{3}h^2$** (approx. $1.33h^2$).
Given the common "identical pieces" geometry standards, the most likely intended answer is **$\frac{5}{2}h^2$** or **$2.5h^2$**.
Answer by ikleyn(53906)  (Show Source):
You can put this solution on YOUR website! .
The large square below is made-up of seven identical rectangles, three identical squares, and six identical triangles.
The height of each of the seven rectangles is h. What is the area of each of the seven rectangles?
(Give your answer in terms of h.)
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After reading this post, I have only one question to a person, who created/posted this text:
How far below is the large square, where is it shown ?
I am asking, because, depending on configuration, different answers are possible.
In other words, generally speaking, the answer depends on configuration.
How the problem is posed in the post, it is not an accurate mathematical formulation.
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