Answer:
35"
Explanation:
Interesting question, and I got to re-remember something from it.
See at the top where they wrote "ΔABC ~ ΔNML"?
The squiggle means the triangles are similar. So they'll have the same angles, and the side lengths will be some multiple of each other.
Now let's figure out which sides correspond to which in these two triangles.
See how the angles ACB and MLN are almost right angles? Let's focus on that particular angle. It doesn't matter what its value is, just that we're looking at the same angle on each triangle. To be painfully clear, I'm looking at the angle next to C, and the one next to L. Got it?
Good, then what are the lengths of the legs of those angles?
For the "C" angle they're 20" and 30".
For "L" they're 32" & 48".
Do you see that? Good, then which side corresponds to which?
Probably the shorter one to the shorter one, and the longer one to the longer one, right?
So 20" corresponds to 32", and 30" corresponds to 48".
Is this making sense? Because now we get to write a "ratio", which is just a comparison of two numbers. Let's start with the corresponding sides 20" and 32". Their ratio is:
20:32 Good so far?
What we need to do is figure out "how much bigger" 32 is than 20. And ratios are like fractions (in fact they are fractions, we just right them sideways by convention) in that you can manipulate them by dividing out common factors, and the ratio stays the same. So divide both sides by 2 to get:
10:16 That's the same ratio as 20:32. See it?
Now hold there for a minute. If this were a fraction we'd want to divide by 2 again and get 5/8 and be done. But for a ratio we want to simplify until one side is "1". <-- super important
How would we do that here? Divide by 10:
1 : 1.6 And that's the ratio we did all that thinking and working for.
It tells us that if we multiply a side length of ΔABC by 1.6 we'll get the corresponding side length of ΔNML. And that works because the two triangles are similar.
That's key, so please take the time to understand it. It's probably in your textbook too, with better examples.
But to convince ourselves that it works, let's test it on the other pair of corresponding legs (30" and 48"): does 30 x 1.6 = 48? It does, so that's a check that we got the ratio right.
Now we're finally able to compute x. Side AB corresponds to side MN, so they're related by the ratio (or factor of) 1.6. But which one is longer, x or 56"? It's x, right? Just look at the other side lengths in comparison. LMN is the bigger triangle. That's important, so you don't set it up backwards and get it wrong after all that work.
So what's the relationship? In words, "56 inches is 1.6 times longer than x." So as an equation:
56 = 1.6x Turn it around and solve for x:
x = 56/1.6 = 35"
I hope that helped, take care.