The base of a triangle is numerically 3 less than its area, and the height is numerically 6 less than its area. Find the area
To find the area of the triangle, we need to know the length of its base and height.
Let's assume the area of the triangle is 'A' square units.
According to the given condition, the base of the triangle is numerically 3 less than its area, so the base will be A - 3.
Similarly, the height of the triangle is numerically 6 less than its area, so the height will be A - 6.
The formula to calculate the area of a triangle is:
Area = (base * height) / 2
Substituting the values we have:
A = ((A - 3) * (A - 6)) / 2
Now, let's solve this equation to find the value of A, which represents the area of the triangle.
Multiplying both sides of the equation by 2:
2A = (A - 3) * (A - 6)
Expanding the equation:
2A = A^2 - 6A - 3A + 18
Simplifying the equation:
2A = A^2 - 9A + 18
Moving all terms to one side:
A^2 - 11A + 18 = 0
Now, we need to solve this quadratic equation for A. Factoring or using the quadratic formula can help us find the values of A. Into the quadratic formula:
A = [-(-11) ± √((-11)^2 - 4*1*18)] / (2*1)
Simplifying further:
A = [11 ± √(121 - 72)] / 2
A = [11 ± √49] / 2
A = [11 ± 7] / 2
This yields two possible values for A:
A = (11 + 7) / 2 = 9
A = (11 - 7) / 2 = 2
Since the area of a triangle cannot be negative, we discard the value A = 2. Thus, the area of the triangle is A = 9 square units.