The height of a triangle is 4 centimeters less than twice the length of its base. If the total area of the triangle is 48 square centimeters, then find the lengths of the base and height.
base ----- x cm
height ---- 2x-4 cm
area of triangle = (1/2) base x height
(1/2)(x)(2x-4) = 48
x(2x-4) = 96
2x^2 - 8x - 96 = 0
x^2 - 4x - 48 = 0
(x-8)(x+6) = 0
x = 8 or x = -6, the latter is inadmissable
base is 8 cm
height is 12 cm
To solve the problem, we can use the formula for the area of a triangle:
Area = (base * height) / 2
Let's denote the base as 'b' and the height as 'h'.
According to the problem, the height of the triangle is 4 centimeters less than twice the length of its base, which can be expressed as:
h = 2b - 4
We also know that the total area of the triangle is 48 square centimeters:
48 = (b * h) / 2
Substituting the expression for height from the first equation into the second equation, we get:
48 = (b * (2b - 4)) / 2
Now we can solve for 'b'.
Multiply both sides of the equation by 2:
96 = b * (2b - 4)
Expand the right side:
96 = 2b^2 - 4b
Rearrange the equation to form a quadratic equation:
2b^2 - 4b - 96 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring is more practical. Let's write it as:
(b - 8)(2b + 12) = 0
This gives two possible solutions for 'b':
b - 8 = 0 or 2b + 12 = 0
If we solve these two equations, we find that:
b = 8 or b = -6
Since we can't have a negative length for the base, we discard the solution b = -6.
Therefore, the length of the base is 8 centimeters.
Now we can calculate the height using the equation:
h = 2b - 4
Substituting the value of base b = 8 into this equation, we get:
h = 2 * 8 - 4
h = 16 - 4
h = 12
Therefore, the length of the height is 12 centimeters.
In summary, the lengths of the base and height of the triangle are 8 centimeters and 12 centimeters, respectively.