The height of a triangle is 6 inches less than the base. The area of the triangle is 67.5 square inches. Find the length of the base and the height of the triangle.
A = bh/2
so,
1/2 b (b-6) = 67.5
Now just solve for b, and h=b-6
To find the length of the base and height of the triangle, we can use the formula for the area of a triangle.
The formula for the area of a triangle is:
Area = (base * height) / 2
Let's use this formula and the given information to solve the problem.
We are given that the height of the triangle is 6 inches less than the base. Let's assume the length of the base is "x" inches. Therefore, the height of the triangle would be "x - 6" inches.
Now, we can substitute these values into the area formula:
67.5 = (x * (x - 6)) / 2
To simplify the equation, let's multiply both sides of the equation by 2:
135 = x * (x - 6)
Expanding the equation:
135 = x^2 - 6x
Rearranging the equation to set it equal to zero:
x^2 - 6x - 135 = 0
Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula. Since this equation doesn't factor easily, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -6, and c = -135. Substituting these values into the quadratic formula, we get:
x = (-(-6) ± √((-6)^2 - 4*1*(-135))) / (2*1)
Simplifying further:
x = (6 ± √(36 + 540)) / 2
x = (6 ± √(576)) / 2
x = (6 ± 24) / 2
Now, we have two possible solutions for x:
1) When x = (6 + 24) / 2 = 30 / 2 = 15
2) When x = (6 - 24) / 2 = -18 / 2 = -9 (but the length can't be negative).
Therefore, the length of the base of the triangle is 15 inches.
Now, let's find the height of the triangle:
Height = base - 6
Height = 15 - 6
Height = 9 inches
So, the length of the base is 15 inches and the height of the triangle is 9 inches.