CLRNC CONFUSION ON TKOF POINT. RWY NOT MARKED ON CLOSED SECTION. RPTR CAPT TOOK OFF ON CLOSED PART OF RWY AND NOTIFIED AFTER TKOF BY TWR.

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AT HOU WE WERE GIVEN THE INSTRUCTIONS TO 'CROSS RWY 12L; TURN L ONTO RWY 17' AND WERE CLRED 'INTO POS AND HOLD RWY 12R.' RWY 12R HAS A CUTBACK OF 1032 FT AND THERE WAS A NOTAM THAT SAID THE FIRST 300 FT OF RWY 12R WAS CLOSED. WE TAXIED ONTO RWY 12R; WAITED APPROX 30 SECONDS; AND THEN WERE CLRED FOR TKOF RWY 12R. AFTER WE WERE AIRBORNE; HOU TWR TOLD US THAT WE HAD USED THE CLOSED 300 FT OF RWY 12R FOR TKOF. HE ALSO STATED THAT HE HAD CLRED US TO TAKE OFF FROM THE INTXN OF RWY 17 AND RWY 12R. THE TAPES WILL SHOW THAT HE DID NOT CLR US FOR TKOF FROM THE INTXN OF RWY 17 AND 12R. I TOLD HIM THAT THE CLOSED PART OF RWY 12R DID NOT HAVE ANY TYPE OF MARKINGS OR BARRICADES TO INDICATE THE CLOSED PART OF THE RWY AND THERE IS NO WAY FOR ME TO DETERMINE WHAT 300 FT SHOULD BE. IT IS VERY OBVIOUS WHERE THE CONSTRUCTION IS AND THE CUTBACK IS AT HOU AND I WAS UNDER THE IMPRESSION THAT THE CUTBACK AND CLOSED PART OF THE RWY WAS PART OF THE OBVIOUS CONSTRUCTION THAT IS TAKING PLACE. SOLUTION: I SUGGEST THAT IF ATC (TWR OR GND) SEES AN OBVIOUS VIOLATION; THAT THEY VERBALIZE THE MISTAKE AND CORRECT IT BEFORE HARM COME TO ANY EQUIP OR PERSONNEL (THE WORKERS ON THE CUTBACK). ALSO; IF A RWY OR TXWY IS CLOSED; THAT SOME TYPE OF MARKINGS OR BARRICADES BE PLACED ON THE CLOSED PART SO MISTAKES CANNOT BE MADE AND HARM COMES TO ANY PERSONNEL OR EQUIP.