During landing of the Cessna 172R, the student pilot reduced the power to begin the flare. The airplane ballooned and the pilot added “a little power” to prevent a hard landing. Once on the runway, the pilot applied the brakes to slow the airplane. The airplane turned left, approximately 45 degrees from center line. The pilot reported he applied full brakes and attempted to return the airplane back toward the runway's centerline, but the airplane did not respond and continued to veer towards the edge of the runway. The airplane departed the left side of the runway, impacting some runway signs and markers. The right wheel strut failed, dropping the airplane's right side and pivoting the airplane's tail. The airplane came to rest on the runway, facing back toward the direction of travel. In the National Transportation Safety Board Form 6120.1 submitted by the pilot, "No" was checked for "Was there a Mechanical Malfunction/Failure."