Similar Triangles - which leg corresponds to which

[NicolasB335]

I don't understand how from the following graph you can determine which leg corresponds to which among the similar triangles.

How do they determine that BC corresponds to CA, and CA corresponds to CD? I can't determine which is the smallest leg between BC and CD, and I can't determine which is the smallest angle between BAC and CAD

Images:

https://ibb.co/8K20qGw
https://ibb.co/jkC9Dvg

[Sage Pearce-Higgins]

Please check the forum guidelines before posting so you post in the appropriate folder - this looks like a non-CAT Manhattan Prep problem.

This problem hinges on the properties of similar triangles. Take a look at the drawing (or, even better, draw it out). Let's call angle BAC y. Then angle CAD will be 90-y. Therefore, since angles in a triangle add up to 180 degrees, angle ADC will be y. Also, angle ABC is 90-y, meaning that triangles ABC and ACD are similar: they have the same angles. Similar triangles have side lengths that are in proportion to each other; one example is a 3-4-5 triangle and a 6-8-10 triangle. You could even cut up your drawing and put the two triangles alongside each other to notice that the shortest side of triangle ABC is 3, and the shortest side of triangle ACD is AC, which is 4. Therefore triangle ACD is bigger than ABC by a factor of 4/3, allowing you to find the other side lengths.