The area of a right triangle is 44m squared. Find the lengths of its legs if one of the legs is 3m longer than the other.
Formula needed is:
A = 1/2(base)(height)
44 = 1/2(x + 3)(x)
One leg is 3m longer than the other can be expressed as (x + 3).
The other leg is simply the letter (x).
In other words, (x + 3) is one of the legs and the letter (x) is the other leg.
This is your equation:
44 = 1/2(x + 3)(x)
Can you solve for x?
If not, write back and I will help you.
(1/2)x(x+3) = 44
x is the shorter of the two legs
x^2 + 3x -88 = 0
That factors easily. (Think 11 x 8)
Take the positive root.
To solve this problem, let's start by assigning variables to the lengths of the legs of the right triangle.
Let's say that one leg of the triangle is x meters long. The problem states that the other leg is 3 meters longer, so the length of the other leg is x + 3 meters.
The area of a right triangle is given by the formula A = 1/2 * base * height, where the base and height of the triangle are the lengths of its legs.
We are given that the area of the right triangle is 44 square meters. So we can set up the equation:
44 = 1/2 * x * (x + 3)
To solve for x, we can simplify the equation:
44 = 1/2 * x^2 + 3/2 * x
Multiplying both sides of the equation by 2 to remove the fraction:
88 = x^2 + 3x
Rearranging the equation to set it equal to zero:
x^2 + 3x - 88 = 0
Now, we can solve this quadratic equation. Since it does not factor nicely, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our quadratic equation, a = 1, b = 3, and c = -88. Plugging these values into the quadratic formula, we have:
x = (-(3) ± √((3)^2 - 4(1)(-88))) / 2(1)
Simplifying inside the square root:
x = (-3 ± √(9 + 352)) / 2
x = (-3 ± √(361)) / 2
We can simplify the square root:
x = (-3 ± 19) / 2
This gives us two possible values for x:
x1 = (-3 + 19) / 2 = 16 / 2 = 8
x2 = (-3 - 19) / 2 = -22 / 2 = -11
Since we are dealing with lengths, we can discard the negative value of x. Therefore, the length of one leg of the right triangle is 8 meters. The length of the other leg is 8 + 3 = 11 meters.
So, the lengths of the legs of the right triangle are 8 meters and 11 meters.
To solve this problem, we can use the formula for the area of a right triangle, which is given by the equation:
Area = (1/2) * base * height
Let's assume that one leg of the right triangle is x meters long. Since we are given that the other leg is 3 meters longer, we can express it as (x + 3).
Given that the area of the right triangle is 44 square meters, we can set up the equation:
44 = (1/2) * x * (x + 3)
To solve this equation, we can multiply both sides by 2 to eliminate the fraction:
88 = x * (x + 3)
Expanding the right side of the equation:
88 = x^2 + 3x
Rearranging the equation to set it equal to zero:
x^2 + 3x - 88 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring might be a bit difficult, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, the coefficients are:
a = 1
b = 3
c = -88
Plugging these values into the quadratic formula:
x = (-3 ± √(3^2 - 4 * 1 * -88)) / (2 * 1)
Simplifying the equation:
x = (-3 ± √(9 + 352)) / 2
x = (-3 ± √361) / 2
Taking the square root of 361:
x = (-3 ± 19) / 2
This gives us two possible solutions for x:
1. x = (-3 + 19) / 2 = 8
2. x = (-3 - 19) / 2 = -11
Since a length cannot be negative, we discard the negative solution. Therefore, the length of one leg of the right triangle is 8 meters.
To find the length of the other leg, we can substitute this value back into (x + 3):
Length of the other leg = 8 + 3 = 11 meters
So, the lengths of the legs of the right triangle are 8 meters and 11 meters, respectively.