- #1

Lancelot1

- 28

- 0

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.

Thank you in advance !

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

What is your conclusion ?

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 ?

View attachment 8008

View attachment 8009

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.

Thank you in advance !

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

What is your conclusion ?

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