MHB Ratio of areas of squares - Challenging problem

Lancelot1
Messages
26
Reaction score
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 ?

My solution (assuming the length is 1):

View attachment 8008

View attachment 8009
 

Attachments

  • square problem 2.PNG
    square problem 2.PNG
    2.4 KB · Views: 123
  • square problem.PNG
    square problem.PNG
    11.9 KB · Views: 126
  • square problem 3.PNG
    square problem 3.PNG
    4.7 KB · Views: 121
  • square problem 4.PNG
    square problem 4.PNG
    8.7 KB · Views: 108
  • square problem 5.PNG
    square problem 5.PNG
    9.7 KB · Views: 125
Mathematics news on Phys.org
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 $S_{KLIJ} = S_{ABCD} - 4S_{\triangle DEA} + 4S_{\triangle DJF}$.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top