Their mother, Christy Sheats, 42, was standing nearby with a gun in her hand, the authorities said.

Image An image of Christy Sheats from her Facebook profile. Credit... YouTube

When the mother refused to drop the weapon, the officer shot her once in the chest, Sheriff Troy E. Nehls said.

On Monday, the sheriff’s office released more details about the shootings, though a motive was not immediately clear. It said Christy Sheats had called a family meeting, and when her husband, Jason Sheats, 45, and their daughters gathered in the living room, the mother opened fire.

The daughters and the father ran out the front door in the 6000 block of Remson Hollow Lane, and Madison Sheats collapsed, the authorities said. Mr. Sheats ran to the end of a cul-de-sac, Taylor Sheats ran into the street, and the mother followed, shooting her.

Christy Sheats went back inside to reload her weapon, returned and shot the older daughter again, according to a witness. Taylor Sheats was taken to a hospital, where she died.

The police found a .38-caliber handgun at the scene, the statement said.

Sheriff Nehls said that crisis intervention teams had been sent to the home more than once in recent years over turmoil that seemed to revolve around the mother.