Let's denote the height of the triangle as 'h' (in centimeters). According to the problem, the base of the triangle is four more than twice its height, which can be expressed as "2h + 4" (in centimeters).
The formula for the area of a triangle is given by:
Area = (1/2) * base * height
Substituting the given values, we have:
54 = (1/2) * (2h + 4) * h
To solve for the height, we can rewrite the equation:
54 = (h/2) * (2h + 4)
Multiplying both sides by 2 to eliminate the fraction:
108 = h * (2h + 4)
Expanding the expression:
108 = 2h^2 + 4h
Rearranging the equation to set it equal to zero:
2h^2 + 4h - 108 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use factoring in this case:
2h^2 + 4h - 108 = (2h - 12)(h + 9) = 0
Setting each factor equal to zero:
2h - 12 = 0 or h + 9 = 0
Solving for 'h' in each case:
2h - 12 = 0
2h = 12
h = 6
h + 9 = 0
h = -9
Since the height of a triangle cannot be negative, we discard the solution 'h = -9'.
Therefore, the height of the triangle is 6 centimeters.
To find the base, substitute the value of height (h) into the expression for the base:
base = 2h + 4
base = 2(6) + 4
base = 12 + 4
base = 16
Hence, the base of the triangle is 16 centimeters.