Similar triangles question

[ggcurly]

Hi. I've been stuck on this question forever (5bii).

Because the pair of triangles are similar, as proven in part A, this means that their sides are proportionate to each other. I gather that I am supposed to compare the ratios of side lengths by using fractions to solve an equation as shown in this example:

But I am confused for two reasons: Firstly, the smaller triangle is positioned differently to the bigger triangle.. they are facing different directions. Secondly, I don't think I have enough information for side lengths to make an equation with only one unknown variable.
Any help appreciated
Thanks so much.

 

[Dr.Peterson]

Orientation has no effect on similarity. What matters is which side correspond, which is indicated by the naming of the figures. Given that CLW and LTE are similar [I've never seen your symbol for similarity, but I take it that's what it means], we know that

CL corresponds to LT​

LW corresponds to TE​

CW corresponds to LE​

Which of these do we know? CL = 15+6 = 21; LT = x, which we want to find; LW = x+5; LE = 6. We don't yet have enough pieces to write a proportion; but we also know they are right triangles, so you could write a couple equations covering the sides we know nothing about yet.

I'd try using that information to write a proportion LT:CL = LE:CW.

On the other hand, it's possible the problem is stated wrong (or I'm misreading your symbol), because the angle at L is shared by the two triangles, so we know that CLW is similar to TLE, and the problem can be more easily solved using that fact. If it is also true that CLW is similar to LTE, that implies a special angle. This may be true and they are trying to fool you, or they may have written the question wrong.