The base of a triangle is 5cm greater than the height. The area is 52cm^2. Find the height and length of the base.
To find the height and length of the base of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
We have been given that the area is 52 cm^2. Plugging this into the formula, we get:
52 = (1/2) * base * height
Now, we are also given that the base of the triangle is 5 cm greater than the height. This can be represented as:
base = height + 5
Substituting this into the equation, we get:
52 = (1/2) * (height + 5) * height
To solve this equation for the height, we can multiply both sides by 2:
104 = (height + 5) * height
Expanding the right side of the equation, we get:
104 = height^2 + 5height
Rearranging the equation to standard quadratic form, we have:
height^2 + 5height - 104 = 0
This is a quadratic equation, and we can solve it by factoring or by using the quadratic formula. In this case, the equation factors as:
(height + 13)(height - 8) = 0
Setting each factor equal to zero, we get two possible solutions:
height + 13 = 0 or height - 8 = 0
Solving these equations, we find that:
height = -13 or height = 8
Since the height of a triangle cannot be negative, we can discard the negative value. Therefore, the height of the triangle is 8 cm.
To find the length of the base, we can substitute the value we found for the height into the equation we set up earlier:
base = height + 5
base = 8 + 5
base = 13
Therefore, the length of the base is 13 cm.
So, the height of the triangle is 8 cm and the length of the base is 13 cm.
area=1/2 b h
52=1/2 (h+5)h
multipy it out, solve for the quadratic
104=h^2+5h
h^2+5h-104=0
use the quadratic equation.