Answer: Third answer choice
Triangle ADB is similar to Triangle CDA
note: the order of the letters matters
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There are many ways to do this. One way that might work is to find the hypotenuse of each triangle. There are 3 triangles shown here which are...
* triangle ABC (smallest triangle on the right side)
* triangle CDA (medium sized triangle on the left)
* triangle ABD (largest triangle overall)
The hypotenuse of each triangle mentioned is
* segment AB (for triangle ABC)
* segment DA (for triangle CDA)
* segment BD (for triangle ABD)
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Now look at the first answer choice. The sequence BDA has BD as the hypotenuse as we're talking about the largest triangle overall. Note how BD is the first two letters of BDA. This means that ACB must have AC as the hypotenuse. But looking at the triangle ABC, the hypotenuse is actually AB. So we can cross choice 1 off the list. As you can see, the order of the letters matters so we can line up the corresponding angles and segments.
Onto the second answer choice. Triangle CAB has hypotenuse AB. We have AB as the last two letters of CAB. The sequence DBA has BA as the last two letters, so BA should be a hypotenuse of triangle BDA; however the hypotenuse is actually BD. Cross choice 2 off the list.
I'm going to skip the third answer choice and come back.
The fourth answer choice is crossed off the list because CDA has AD as the hypotenuse (last two letters) so for ABC, the hypotenuse must be BC (last two letters). But ABC has a hypotenuse of AB.
Coming back to the third answer choice. We have DB as the hypotenuse of ADB. Similarly, DA is the hypotenuse of CDA. both DB and DA are the last two letters of the sequences ADB and CDA. You'll find that the short legs and long legs also match up as well. It's just easier to compare hypotenuses.
This confirms that the answer is choice 3.
