The base of a triangle is 4cm greater than the height the area is 48cm^2 find the height and the length of the base
Let the height be x cm
so the base is x+4 cm
Area of triangle = (1/2)base x height
so 48 = (1/2)(x)(x+4)
96 = x^2 + 4x
x^2 + 4x - 96 = 0
Can you solve this quadratic?
Hint: it factors quite nicely
2x-3
Let's assume the base of the triangle is represented by "b" centimeters and the height is represented by "h" centimeters.
According to the given information, the base of the triangle (b) is 4cm greater than the height (h). In equation form, this can be written as:
b = h + 4
The area of a triangle is given by the formula:
Area = 1/2 * base * height
We are given that the area is 48cm^2, so we have:
48 = 1/2 * b * h
Substituting the value of b from the first equation, we have:
48 = 1/2 * (h + 4) * h
Expanding and simplifying:
48 = 1/2 * (h^2 + 4h)
Multiplying both sides of the equation by 2 to eliminate the fraction:
96 = h^2 + 4h
Rearranging the equation into standard quadratic form:
h^2 + 4h - 96 = 0
Now, we can solve this quadratic equation to find the value of h using factoring, completing the square, or quadratic formula. Let's use factoring:
(h + 12)(h - 8) = 0
Setting each factor equal to zero:
h + 12 = 0 or h - 8 = 0
Solving each equation separately:
h = -12 or h = 8
Since the height cannot be negative, we discard the solution h = -12.
Therefore, the height of the triangle is 8 cm.
To find the base (b), we substitute the value of h in the first equation:
b = h + 4 = 8 + 4 = 12
Therefore, the length of the base is 12 cm.
To find the height and length of the base of the triangle, you can use the formula for the area of a triangle, which is:
Area = (1/2) × base × height
Given that the area is equal to 48 cm^2, we can substitute this into the formula and solve for the height.
48 = (1/2) × base × height
Next, we are told that the base is 4 cm greater than the height. We can express this as:
base = height + 4
Substituting this expression for the base in the area formula, we get:
48 = (1/2) × (height + 4) × height
Now, we can simplify and solve for the height. Distribute the 1/2 factor and solve the quadratic equation:
48 = (1/2) × (height^2 + 4height)
96 = height^2 + 4height
0 = height^2 + 4height - 96
This quadratic equation can be factored as follows:
0 = (height - 6)(height + 16)
Setting each factor equal to zero and solving for the height gives us two possible solutions:
(height - 6) = 0 --> height = 6
(height + 16) = 0 --> height = -16
Since the height of a triangle cannot be negative, we discard the second solution.
Therefore, the height of the triangle is 6 cm.
To find the length of the base, substitute this height value into the expression for the base:
base = height + 4
base = 6 + 4
base = 10 cm
So, the height of the triangle is 6 cm and the length of the base is 10 cm.