For this scalene triangle: which of the following is true?

[s3a]

Homework Statement

The problem is attached as TheProblem.jpg and the answer is D.

Homework Equations

Perhaps the equations attached in Metric relationships summary.pdf.

The Attempt at a Solution

My logic is as follows:
For A, XW does not necessarily equal WZ.

For B, it seems that they are trying to trick the student into thinking of h^2 = d*e (from the Metric relationships summary.pdf file).

For C, the ratio does not work out because the angle facing each respective smaller/larger leg is different.

For D, I can't figure it out but it seems to relate to the Metric relationships summary.pdf file I attached.

Any input would be greatly appreciated!
Thanks in advance!

 

[Simon Bridge]

The metric summary table does not quite apply here because the line YW is not perpendicular to XZ.

What it shows you is that the angle at Y has been bisected - so that ∠XYW = ∠WYZ.
The question wants to know if you understand what this does to the relationships.

One way of thinking this through is to figure wat it would take for each condition to be true: what does it mean?

For instance, A is true if the two sub-triangles are similar (since they share a side, this means the overall triangle XYZ must be isosceles.)

Of course, another approach is to physically (and carefully) draw a few (large) scalene triangles, bisect one angle, and then measure the corresponding sides :)

 

[s3a]

Thanks!

Thanks to you telling me to focus on the bisected angle, I basically converted the answer D into words to make sense of it:

(small leg of small triangle)/(hypotenuse of small triangle) = (small leg of large triangle)/(hypotenuse of large triangle).

The "hypotenuse of small triangle" part is technically not the hypotenuse of the small triangle but rather a length that is equivalently large as it.

 

[Simon Bridge]

Great, well done.
Putting it in words is a pain - personally I relabel these things with lower case for side lengths and upper case for the corresponding angles.