This car has a timing chain. One day my Corolla wouldn't start, but it would turn over. It sounded a little bit different when turning over. I suspected the fuel pump or the well-worn ignition switch, but it turned out to be the starter. I took the starter to two or three auto parts stores to have it tested. All three places told me it "passed" the test, but one (or two?) of the places made a comment saying that although it passed, it was a "weak" pass. That was enough for me, so I replaced it and that solved the problem. That was at a lot more miles than you have, though (281K). Anyway, in addition to the things New Echo told you, you can also check the starter.
By the way, I first bought a rebuilt starter from Advance Auto Parts. It was $99, but with the coupon I got it for $60. It only lasted about 3K or 4K miles. Piece of junk. I then bought a NEW TYC starter online. Much better quality and I recommend that brand if it turns out your problem is the starter.