03/17/2021 (2011: Countdown Round, Problem 23)

Q: If triangle PNM is a right triangle, as shown with PM = 5 units, PN = 4 units, and NQ ⊥ PM, what is the length of segment NQ, in units? Express your answer as a decimal to the nearest tenth.

A: 12/5

 First, label the side lengths:

 

Next, using the Pythagorean theorem, we find MN:

 

Let's now find the angle measures in terms of a, where a = angle P

 

Now, we can see that PMN ~ (is similar to)  PNQ. We can prove this using Angle-Angle Similarity. That means: PM/MN = PN/QN 

We can plug in the values we know to get: 5/3 = 4/QN, which means that QN = 12/5