A right triangle has a base that is 2 more than twice the height. Find the length of the base for the triangle. If the area is 30 square units.
1/2 b h=30
1/2(2x+2)(x)=30
x^2+x-30=0
(x+6)(x-5)
x=-6 Does not make sense
x=5 2x5 +2 =12
Base 12
height 5
To find the length of the base of the right triangle, we can set up an equation using the given information.
Let's assume the height of the right triangle is represented by 'h'. According to the problem, the base is 2 more than twice the height. Therefore, the base can be represented as '2h + 2'.
The formula for the area of a right triangle is given by:
Area = (base * height) / 2
We are given that the area is 30 square units. Substituting the values into the formula, we get:
30 = ((2h + 2) * h) / 2
Simplifying the equation, we have:
30 = (2h^2 + 2h) / 2
Multiplying both sides of the equation by 2, we get:
60 = 2h^2 + 2h
Rearranging the equation to standard quadratic form, we have:
2h^2 + 2h - 60 = 0
To solve this quadratic equation, we can either use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 2, b = 2, and c = -60. Substituting these values into the formula, we get:
h = (-2 ± √(2^2 - 4 * 2 * -60)) / (2 * 2)
Simplifying further, we have:
h = (-2 ± √(4 + 480)) / 4
h = (-2 ± √484) / 4
h = (-2 ± 22) / 4
Simplifying the two possible values for h, we have:
h₁ = (-2 + 22) / 4 = 20 / 4 = 5
h₂ = (-2 - 22) / 4 = -24 / 4 = -6
Since we are dealing with measurements of a triangle, we can disregard the negative value for the height. Therefore, the height 'h' is 5 units.
Now, to find the length of the base, we substitute the value of 'h' back into the equation for the base:
base = 2h + 2
base = 2 * 5 + 2
base = 10 + 2
base = 12
Therefore, the length of the base for the right triangle is 12 units.