Richard Langley celebrates his goal with Rob Earnshaw
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Richard Langley improved Cardiff's chances of reaching the play-offs with the winner at Bradford.
Cardiff were reduced to 10-men just after the hour mark when Gary Croft was dismissed for a foul on Danny Cadamarteri.
But Bradford could not make the most of their numerical advantage and conceded a penalty when Simon Francis fouled
Tony Vidmar.
Robert Earnshaw's spot-kick was saved by Alan Combe but Langley headed in.
Wales striker Earnshaw did not take long to pose a threat and flashed a shot wide in the 12th minute.
Andy Gray looked dangerous for the home side and volleyed wide from 25 yards before a curling right-foot shot from a similar distance was saved by keeper Martyn Margetson.
Both sides were struggling to create chances or get their efforts on target when given a sight goal.
Bradford keeper Combe was rarely troubled, although he had to be alert to hold Peter
Thorne's 20-yard effort.
The deadlock was finally broken after 72 minutes when Francis fouled Vidmar in the box.
Earnshaw's spot-kick was parried by Combe and Langley
was first to the rebound to head home.
Bradford: Combe, Francis, Wetherall, Gavin, Heckingbottom, Summerbee, Atherton, Gray, Armstrong, Cadamarteri, Branch. Subs: Davies, Bower, Emanuel, Muirhead, Standing.
Cardiff: Margetson, Weston, Vidmar, Gabbidon, Croft, Langley, Boland, Kavanagh, Parry, Thorne, Earnshaw. Subs: Campbell, Lee, Barker, Whalley, Alexander.
Referee: C Webster (Tyne & Wear).