Two sides of a triangle have lengths 7 and 12. Which of the following is a possible length for the third side?

By the triangle inequality, the third side $x$ must be greater than the difference of the other two and less than their sum: $12 - 7 < x < 12 + 7$, so $5 < x < 19$. Only 15 lies strictly in that range, choice D. Choice A (3) is too short: $7 + 3 = 10 < 12$, so the sides cannot close up. Choice B (19) equals the sum $7 + 12$, which makes a degenerate triangle flattened into a line. Choice C (20) exceeds the sum, so the two shorter sides cannot reach each other. The inequality must be strict at both ends.