Confusion About A Question Involving Triangles

Question

I've come across a question that confuses me greatly:

I am very confused with the explanation of this question:

How can we assume that $2l=24$? Why can we assume that $2l=24$ after the fact that we assume it's a right triangle with height 24 and a hypotenuse of 2l?

I don't get the next step either. How can we assume that the proportions that hold for the hypotenuse also hold for the height, and therefore deduce that$w=(1/4)12=3?$

Can someone explain this to me, thanks?

Answer 1

IMO, the explanation is not very clear...

We have three similar triangles, that give us two proportions :

$\dfrac {24}{2l} = \dfrac x l$,

and :

$\dfrac {24}{2l} = \dfrac {x+w}{\frac 5 4 l}$,

where $x$ is the height of the topmost right triangle .

This amounts to the following equations :

$x= 12$ and $x+w = 15$,

from which :

$w=3$.

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The author has "squeezed" them into the following :

$24 : 2l = x : l = (x+w) : \dfrac 5 4 l$.

Again, the first two give : $x=12$ (and not $l$) and the last two give : $(12+w)=\dfrac 5 4 12$.

Answer 2

Let $w=\frac{1}{4}x$.

Thus, by Thales $$x+\frac{1}{4}x+\frac{3}{4}x=24.$$ Can you end it now?

I also got $w=3.$