Answer:
Let's call the length of the base "b".
Since the height is 12 cm longer than the base, we can write that the height is equal to "b + 12".
We know the area of the triangle, so we can use the formula for the area of a triangle:
Area = (1/2)bh
Substituting the expressions we have for the base and the height:
270 = (1/2)(b)(b + 12)
Expanding the right side of the equation:
270 = (1/2)b^2 + (1/2)(12b)
Multiplying both sides of the equation by 2:
540 = b^2 + 12b
Combining the terms on the right side of the equation:
540 = b^2 + 12b
Subtracting 12b from both sides of the equation:
540 - 12b = b^2
Isolating b^2 on one side of the equation:
b^2 = 540 - 12b
Moving all the terms to the left side of the equation:
b^2 - 12b + 540 = 0
We can now use the quadratic formula to solve for b:
b = (-(-12) ± √((-12)^2 - 4(1)(540))) / 2(1)
Simplifying:
b = (12 ± √(144 + 2160)) / 2
b = (12 ± √(2304)) / 2
b = (12 ± 48) / 2
The two possible values of b are:
b = (12 + 48) / 2 = 30
b = (12 - 48) / 2 = -18
Since the length of the base cannot be negative, the only possible value for b is 30.
The height of the triangle can be found by using the expression "b + 12":
h = b + 12 = 30 + 12 = 42
So the base of the triangle is 30 cm and the height is 42 cm.