How can I find the ratio between these two sides of a rectangle?

[ericasong]

Kabir divides a rectangle ABCD into eight non-overlapping squares as shown below. Find the ratio of AB to AD.

I started with the inner most square and called that side [MATH]a_1[/MATH]. Then the sides of the second biggest square I called [MATH]a_2[/MATH]. Then

[MATH]a_1+a_2=a_3[/MATH]
[MATH]a_1+a_3=a_4[/MATH]
[MATH]a_1+a_4=a_5[/MATH]
Just to clarify, [MATH]a_5[/MATH] is the top left square and [MATH]a_4[/MATH]is the top right square.

[MATH]a_5+a_1+a_3= DC[/MATH]
Since it's a rectangle, [MATH]AB=DC=a_5+a_3+a_1[/MATH]


However here I don't know how to proceed or if I have been attempting this correctly.
[MATH] [/MATH]

 

[JeffM]

Your work may very well be reasonable, but, if you want someone to follow your logic, I strongly suggest that you give a picture with the squares labelled

[MATH]A_1, \ A_2 \ ... A_7,\ A_8[/MATH]
As it is, you seem at first to be implying that your squares are labelled in ascending order by area.

If that is what you meant, it is false that

[MATH]a_1 + a_2 = a_3.[/MATH]
As I said, I suspect your approach or one similar will work, but please make it easier to follow.

 

[lev888]

Good start. You need to continue expressing larger segment lengths through smaller ones. If you know what a₃ is in terms of smaller segments, then use smaller lengths in subsequent expressions. You should probably arrive at something like this: AB = 5a₁, AD = 7a₁ (just an example), which will allow you to determine the ratio.