Answer: B. $18$

Explanation:1. **Recall the formula for the area of a triangle:** Area ($A$) = $\dfrac{1}{2} \times \text{base} \times \text{height}$ $A = \dfrac{1}{2}bh$

2. **Define variables based on the problem statement:** Let $h$ be the height of the triangle in centimeters. The base ($b$) is $12$ centimeters greater than the height, so $b = h + 12$. The area ($A$) is $270$ square centimeters.

3. **Substitute the expressions into the area formula:** $270 = \dfrac{1}{2}(h+12)(h)$

4. **Solve the equation for $h$:** Multiply both sides by $2$: $540 = (h+12)h$ $540 = h^2 + 12h$ Rearrange into a standard quadratic equation form ($ax^2+bx+c=0$): $h^2 + 12h - 540 = 0$

We can solve this by factoring or using the quadratic formula. Let's try factoring. We need two numbers that multiply to $-540$ and add to $12$. Consider factors of $540$. Since the sum is positive and product is negative, one number is positive and the other is negative, with the positive one having a larger absolute value. Pairs: $(1, 540), (2, 270), (3, 180), (4, 135), (5, 108), (6, 90), (9, 60), (10, 54), (12, 45), (15, 36), (18, 30), (20, 27)$. We need a difference of $12$. The pair $(18, 30)$ has a difference of $12$. We need $+12h$, so we use $+30$ and $-18$. $(h+30)(h-18) = 0$ This gives two possible solutions for $h$: $h+30 = 0 \Rightarrow h = -30$ $h-18 = 0 \Rightarrow h = 18$

5. **Choose the valid solution for height:** Since height must be a positive length, $h = 18$ cm.

6. **Conclusion:** The height of the triangle is $18$ centimeters.