The base of a triangle is 3cm grater than the height. The area is 14cm^2. Find the height and length of the base. Please show how you dervied at your answer.
base = height + 3 cm
Area
= bh/2
= (h+3)h/2
14 = (h+3)h/2
Solve for h to get h=4 and b=7
To find the height and the length of the base of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Given that the area is 14 cm^2, we can substitute the known values into the equation:
14 = (1/2) * base * height
Since we know that the base is 3 cm greater than the height, we can substitute height + 3 for the base:
14 = (1/2) * (height + 3) * height
Next, we can simplify the equation by multiplying both sides by 2 to eliminate the fraction:
28 = (height + 3) * height
Expanding the equation:
28 = height^2 + 3height
Now, let's rearrange the equation to get it in standard quadratic form, where one side is equal to zero:
height^2 + 3height - 28 = 0
This equation is a quadratic equation that can be factored or solved using the quadratic formula. I will use factoring to solve it:
(height + 7)(height - 4) = 0
Setting each factor equal to zero:
height + 7 = 0 or height - 4 = 0
Solving for height:
height = -7 or height = 4
Since the height of a triangle cannot be negative, we discard the negative value, leaving us with:
height = 4 cm
Now, substitute this value back into the equation to find the length of the base:
base = height + 3
base = 4 + 3
base = 7 cm
Therefore, the height of the triangle is 4 cm and the base is 7 cm.