In Louis Sachar's novel, Holes, why did Kate Barlow shoot the sheriff?
To answer the question, one needs to scrutinise the context in which the event occurred.
Kate was the local school teacher and much loved by the children. She fell in love with Sam, 'the onion man', so called because he grew onions across the lake which he would come to sell in town. His onions were very popular since they were believed to have healing properties. He regularly did favours for Miss Katherine (as she was called then) and fixed up the schoolhouse. Whatever task she gave him, he would always assert, 'I can fix that.'
Charles Walker (aka Trout), Sam's competition, had an eye on Miss Katherine and would attend her evening classes just to get her attention, but she was not interested in him even though he was the son of the richest man in the county. Trout was loud, arrogant and stupid.
When one of the residents in the town (Hettie Parker) saw Miss Katherine and Sam kiss, the news spread like wildfire through the town. The townspeople did not allow their children to attend school the next morning. In its place was a lynch mob led by Trout Walker. They came to destroy the schoolhouse. The townspeople were clearly racist and found it unacceptable that Ms Katherine should have a relationship with someone of a different racial persuasion.
Ms Katherine ran to the sheriff, begging for help: "They're destroying the schoolhouse," she said, gasping for breath. "They'll burn it to the ground if someone doesn't stop them!"
"Just calm your pretty self down a second," the sheriff said in a slow drawl. "And tell me what you're talking about."
He got up from his desk and walked over to her.
"Trout Walker has— "
"Now don't go saying nothing bad about Charles Walker," said the sheriff.
"We don't have much time!" urged Katherine. "You've got to stop them."
"You're sure pretty," said the sheriff.
Miss Katherine stared at him in horror.
"Kiss me," said the sheriff.
She slapped him across the face.
He laughed. "You kissed the onion picker. Why won't you kiss me?"
She tried to slap him again, but he caught her by the hand.
She tried to wriggle free. "You're drunk!" she yelled.
I always get drunk before a hanging."
"A hanging? Who— "
"It's against the law for a Negro to kiss a white woman."
"Well, then you'll have to hang me, too," said Katherine. "Because I kissed him back."
"It ain't against the law for you to kiss him," the sheriff explained. "Just for him to kiss you."
"We're all equal under the eyes of God," she declared.
The sheriff laughed. "Then if Sam and I are equal, why won't you kiss me?"
He laughed again. "I'll make you a deal. One sweet kiss, and I won't hang your boyfriend.
I'll just run him out of town."
Miss Katherine jerked her hand free. As she hurried to the door, she heard the sheriff say, "The law will punish Sam. And God will punish you."
She stepped back into the street and saw smoke rising from the schoolhouse.
The extract clearly shows that the sheriff was not intent on doing his job. In fact, it seems as if he was in cahoots with the rest of the townsfolk. The eventual outcome was that Miss Katherine warned Sam and the two of them tried to escape in his boat by crossing the river. Unfortunately, they were run down by Trout in his father's much faster boat. Sam was shot and killed in the water and when Miss Katherine returned to the shore, she found that Sam's beloved donkey, Mary Lou, had been shot in the head.
Three days later Miss Katherine shot the sheriff and gave him the kiss he had asked for. She killed him to avenge the murder of her dearest love. The sheriff had been complicit in his killing for he did not lift a finger to help. He had abandoned his duty because of his prejudice and the obviously corrupt relationship he had had with the Walkers.