A fiel is in the shape of a right triangle. The fence around the perimeter of the field measures 40 m. If the length of the hypotenuse is 17 m, what will be the length of other two sides.
To find the lengths of the other two sides of the right triangle, we can use the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's call the two legs (sides) of the triangle "a" and "b". The hypotenuse "c" is given as 17 m. The perimeter of the field is 40 m, so we have:
a + b + c = 40
Since c is the hypotenuse, we can write:
a + b + 17 = 40
Now, we have two equations:
1) a^2 + b^2 = c^2 (from the Pythagorean theorem)
2) a + b + 17 = 40
We can subtract 17 from both sides of equation 2, giving us:
a + b = 40 - 17
a + b = 23
Now, we have a system of two equations:
a^2 + b^2 = c^2
a + b = 23
To solve this system, we can use substitution or elimination. Let's use substitution.
From equation 2, we can isolate a:
a = 23 - b
Substituting this value of a into equation 1, we have:
(23 - b)^2 + b^2 = 17^2
Expanding and simplifying:
529 - 46b + b^2 + b^2 = 289
Combining like terms:
2b^2 - 46b + 240 = 0
We can simplify this quadratic equation by dividing both sides by 2:
b^2 - 23b + 120 = 0
Now we can factor the equation:
(b - 8)(b - 15) = 0
Setting each factor equal to zero gives us two possible values of b:
b - 8 = 0 -> b = 8
b - 15 = 0 -> b = 15
So, the two possible lengths for side b are 8 m and 15 m.
To find the corresponding values of side a, we can substitute these values of b into equation 2:
For b = 8: a + 8 = 23 -> a = 23 - 8 = 15
For b = 15: a + 15 = 23 -> a = 23 - 15 = 8
Therefore, the possible lengths of the other two sides are:
a = 15 m, b = 8 m
a = 8 m, b = 15 m
To find the lengths of the other two sides of the right triangle, we can use the perimeter and the length of the hypotenuse.
Let's assume that the two shorter sides of the triangle are represented by 'a' and 'b'. And the hypotenuse is represented by 'c'.
We know that the perimeter of a triangle is calculated by adding the lengths of all three sides. So, in this case, we have:
Perimeter = a + b + c = 40 meters
And we are also given that the length of the hypotenuse 'c' is 17 meters.
Now, we can substitute the values into the equation:
a + b + 17 = 40
To solve for 'a' and 'b', we need to isolate them on one side of the equation. So, we subtract 17 from both sides:
a + b = 40 - 17
a + b = 23
Now, we have a system of equations with two variables. Since the triangle is a right triangle, we can use the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Using the Pythagorean theorem, we have:
a^2 + b^2 = c^2
Substituting the values we know:
a^2 + b^2 = 17^2
a^2 + b^2 = 289
Now, we have a system of equations:
a + b = 23
a^2 + b^2 = 289
There are several methods to find the values of 'a' and 'b' from this system of equations, such as substitution or elimination. I will use the substitution method:
Rearranging the first equation, we have:
a = 23 - b
Now, substitute this value of 'a' into the second equation:
(23 - b)^2 + b^2 = 289
529 - 46b + b^2 + b^2 = 289
2b^2 - 46b + 240 = 0
Now, we solve this quadratic equation to find the values of 'b'. Using factoring or the quadratic formula, we find that 'b' can be either 6 or 20.
If 'b' is 6, then substituting this value into the first equation, we find 'a' to be 17.
If 'b' is 20, then substituting this value into the first equation, we find 'a' to be 3.
Therefore, the lengths of the other two sides of the triangle are either 17 meters and 6 meters, or 3 meters and 20 meters.
hmm. 8-15-17 is a standard right triangle, and p=40. Ta-daah!
algebraically,
x^2 + (40-17-x)^2 = 17^2
x = 8 or 15