Choice B is correct. The area, $A$ , of a triangle is given by the formula $A = \frac{1}{2} b h$ , where $b$ represents the length of the base of the triangle and $h$ represents its height. It’s given that the area of a triangle is $270$ square centimeters and that the length of the base of this triangle is $12$ centimeters greater than the height of the triangle. Let $x$ represent the height, in centimeters, of the triangle. It follows that the length of the base of the triangle can be expressed as $x plus 12$ . Substituting $270$ for $A$ , $x$ for $h$ , and $x plus 12$ for $b$ in the formula $A = \frac{1}{2} b h$ yields $270 = \frac{1}{2} (x + 12) (x)$ , or $270 = \frac{1}{2} x (x + 12)$ . Multiplying both sides of this equation by $2$ yields $540 = x (x + 12)$ . Applying the distributive property on the right-hand side of this equation yields $540 = x^{2} + 12 x$ . Subtracting $540$ from both sides of this equation yields $0 = x^{2} + 12 x - 540$ . In factored form, this equation is equivalent to $(x + 30) (x - 18) = 0$ . Applying the zero product property, it follows that $x + 30 = 0$ or $x - 18 = 0$ . Subtracting $30$ from both sides of the equation $x + 30 = 0$ yields $x = - 30$ . Adding $18$ to both sides of the equation $x - 18 = 0$ yields $x equals 18$ . Since $x$ represents the height of the triangle, it must be positive. Therefore, the height, in centimeters, of the triangle is $18$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.