A garden has the shape of a right triangle with one leg 3 meters longer tha the other. The hypotenuse is 3 meters less than twice the lenght of the shorter leg.
b = a + 3
c = 2a -3
c^2 = a^2 + b^2 (Pythagorean theorem)
Solve those three equations simultaneously.
c^2 = 4a^2 -12a +9
c^2 = a^2 + (a+3)^2 = 2a^2 +6a +9
2a^2 -18a =0
a = 9
b = 12
c = 15
s^2 + l^2 = h^2
s^2 + (s+3)^2 = (2s-3)^2
s^2 + s^2 + 6 s + 9 = 4 s^2 - 12 s + 9
2 s^2 - 18 s = 0 = 2s(s-9)
s = 0 or s = 9
the s = 0 root is irrelevant since that results in a negative hypotenuse
s = 9
l = 9+3 = 12
h = 15
To solve this problem, let's assign variables to the lengths of the legs of the triangle.
Let's say the length of the shorter leg is x meters.
According to the problem, the longer leg is 3 meters longer than the shorter leg. So, the length of the longer leg can be expressed as (x + 3) meters.
The hypotenuse is 3 meters less than twice the length of the shorter leg. Therefore, the length of the hypotenuse can be expressed as (2x - 3) meters.
Now, we can use the Pythagorean theorem to find the value of x.
According to the Pythagorean theorem, the sum of the squares of the two legs is equal to the square of the hypotenuse.
So, we have the equation:
x^2 + (x + 3)^2 = (2x - 3)^2
Expanding and simplifying the equation:
x^2 + x^2 + 6x + 9 = 4x^2 - 12x + 9
Rearranging the terms:
2x^2 + 6x = 4x^2 - 12x
Subtracting 2x^2 and 6x from both sides:
0 = 2x^2 - 18x
Dividing both sides by 2x (assuming x is not equal to zero):
0 = x - 9
So, x = 9.
Therefore, the length of the shorter leg is 9 meters, and the longer leg is (9 + 3) = 12 meters. The hypotenuse is (2 * 9 - 3) = 15 meters.
Thus, the sides of the right triangle are 9 meters, 12 meters, and 15 meters.
To solve this problem, let's assign variables to the lengths of the two legs. Let's say that the shorter leg has a length of x meters. According to the problem, the other leg is 3 meters longer, so its length would be x + 3 meters.
Now, let's consider the hypotenuse. The problem states that the hypotenuse is 3 meters less than twice the length of the shorter leg. Therefore, the hypotenuse can be represented as 2x - 3 meters.
Using the Pythagorean theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b), we can set up the equation as follows:
x^2 + (x + 3)^2 = (2x - 3)^2
Simplifying this equation will allow us to solve for x, the length of the shorter leg.
x^2 + (x^2 + 6x + 9) = (4x^2 - 12x + 9)
Combine like terms:
2x^2 + 6x + 9 = 4x^2 - 12x + 9
Rearrange the equation, bringing all the terms to one side:
2x^2 - 4x^2 + 6x + 12x = 9 - 9
-2x^2 + 18x = 0
Factor out the common term "x":
x(-2x + 18) = 0
Now, we have two possible solutions:
1) x = 0 (This doesn't make sense in the context of the problem since the length of a side cannot be zero).
2) -2x + 18 = 0
Solving the second equation for x:
-2x + 18 = 0
-2x = -18
x = -18 / -2
x = 9
Therefore, the length of the shorter leg is 9 meters. The length of the longer leg is 9 + 3 = 12 meters, and the length of the hypotenuse is 2(9) - 3 = 15 meters.