# Ratio of areas of squares - Challenging problem

• MHB
• Lancelot1
In summary, the conversation discusses a difficult geometry problem involving a square and similar triangles. The first question asks for the ratio of areas between two squares, while the second question changes the given ratio. The third question asks for a conclusion and the fourth question involves analyzing a graph and its function's behavior. The conversation also includes a solution to the problem using similar triangles and an alternative solution using areas.
Lancelot1
Hello all,

I have encountered a very difficult question in geometry. The question has several parts. I really need your help. I have tried solving the first and second ones, not sure I did it correctly, and certainly don't know how to proceed and what the results means. I would really appreciate your help in solving this tricky one...My solution is at the end, below the question.

1) A square ABCD is given. Each vertex is connected with a point on the opposite edge (clockwise) such that the ratio between the closer part to the vertex and the edge of the square is 1:3. Find the ratio of areas between the squares KLIJ and ABCD.

View attachment 8006

2) Solve the previous problem when the ratio is 1:4 instead of 1:3.

3) Complete the following table:

View attachment 8005

4) Look at the graph of

$f(x)=\frac{(x-1)^{2}}{x^{2}+1}$

View attachment 8007

What is the geometric explanation to the function's behavior ?
What is the meaning of area where the function increases / decreases ? What is the meaning of the asymptote ? Can this function be generalized to the negative region ? What does it mean ?

My solution (assuming the length is 1):

View attachment 8008

View attachment 8009

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This is an interesting problem. I found it easier to take the size of the large square $ABCD$ to be $n\times n$, so that the lengths of the segments $AE$, $BH$, ... , are $1$.

I think that the best way to tackle the problem is to use similar triangles. The triangles $ABH$ and $AIE$ are similar (having the same angles). The sides of the larger triangle $ABH$ are $1$, $n$ and (by Pythagoras) $\sqrt{n^2+1}$. The hypotenuse of the smaller triangle $AIE$ is $1$, so the ratio of corresponding sides in the triangles is $1:\sqrt{n^2+1}$. From that, you can find the lengths of the other two sides of the smaller triangle.

Now look at the line $AH = AI + IL + LH$. In that equation, you now know everything except $IL$. So you can use that to find the length of the sides of the inner square $IJKL$, which turns out to be $\dfrac{n(n-1)}{\sqrt{n^2+1}}.$ That gives the ratio of the areas of the squares $ABCD$ and $IJKL$ to be $\dfrac{(n-1)^2}{n^2+1}$.

Your calculation using areas is also a good method, and it is correct right up to the last line $S_{KLIJ} = S_{ABCD} - 3S_{\triangle DEA} + 4S_{\triangle DJF}$, which should be $S_{KLIJ} = S_{ABCD} - 4S_{\triangle DEA} + 4S_{\triangle DJF}$.

## 1. What is the ratio of the areas of two squares?

The ratio of the areas of two squares is equal to the ratio of their side lengths squared. This means that if the side length of one square is twice the length of the other, the ratio of their areas will be 4:1.

## 2. How is the ratio of areas of squares related to their side lengths?

The ratio of areas of squares is directly proportional to the square of their side lengths. This means that as the side length of a square increases, the ratio of its area to another square will also increase.

## 3. Can the ratio of areas of squares be greater than 1?

Yes, the ratio of areas of squares can be greater than 1. This happens when the side length of one square is greater than the side length of the other square, resulting in a larger area.

## 4. How can I use the ratio of areas of squares to solve a challenging problem?

The ratio of areas of squares can be used to compare the size of different squares and solve problems involving geometric or real-life scenarios. By understanding the relationship between side lengths and areas, you can use this ratio to find missing values or make predictions.

## 5. Is the ratio of areas of squares always the same for all squares?

No, the ratio of areas of squares will vary depending on the size of the squares. However, the ratio will always follow the same pattern of being equal to the ratio of their side lengths squared.

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