suppose a triangle has an area of 10ft^2. IF the base of the triangle is 1 foot longer than the height of the triangle, what are the lenghts of the base and height?
please help and show work so i understand how to do it
To solve this problem, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Given that the area of the triangle is 10 ft^2, we can plug in this value and symbolically represent the base and height as follows:
10 = (1/2) * base * height
Since the base is given as 1 foot longer than the height, we can express it as:
base = height + 1
Now we can substitute this expression for the base in the equation:
10 = (1/2) * (height + 1) * height
To simplify further, we can distribute the (1/2) to the terms inside the parentheses:
10 = (1/2) * (height^2 + height)
Next, we can eliminate the fraction by multiplying both sides of the equation by 2:
20 = height^2 + height
Rearranging the equation to the standard quadratic form:
height^2 + height - 20 = 0
Now we can factor this quadratic equation:
(height + 5)(height - 4) = 0
Setting each factor equal to zero:
height + 5 = 0 --> height = -5
height - 4 = 0 --> height = 4
Since the height cannot be negative, we discard the solution height = -5.
So, the height of the triangle is 4 ft. Since the base is 1 foot longer, the base length is:
base = height + 1 = 4 + 1 = 5 ft
Therefore, the lengths of the base and height are 5 ft and 4 ft, respectively.