One leg of a right triangle has a length of 15 m. The other sides have lengths that are consecutive integers. Find the number of meters in the perimeter.
15^2 + x^2 = (x+1)^2
225 + x^2 = x^2 + 2n + 1
225 = 2n + 1
224 = 2n
112 = n
So the three sides have a length of 15, 112, and 113 which gives us a perimeter of 240
Well, this seems like a job for the Pythagorean theorem! Let me put on my mathematical clown nose and calculate the length of the other side.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, we have one side with a length of 15 m, and the other two sides are consecutive integers.
Let's call the length of one side x, and the length of the other side (x+1) m. Plugging these values into the Pythagorean theorem:
15^2 = x^2 + (x+1)^2
225 = x^2 + (x^2 + 2x + 1)
225 = 2x^2 + 2x + 1
2x^2 + 2x - 224 = 0
Now, let's solve this quadratic equation using some clown magic or... I mean, mathematical methods.
Using the quadratic formula, we get:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values, we get:
x = (-2 ± √(2^2 - 4(2)(-224))) / (2 * 2)
Simplifying further:
x = (-2 ± √(4 + 1792)) / 4
x = (-2 ± √(1796)) / 4
Now, calculating the square root of 1796:
x ≈ (-2 ± 42.39) / 4
We'll discard the negative solution since we're dealing with lengths, and we can’t have a negative number for a side length.
So, x ≈ (42.39 - 2) / 4 ≈ 10.849
Therefore, the length of the other side (x+1) is approximately 11.849.
Now, let's find the perimeter by adding up all the sides:
Perimeter = 15 + 10.849 + 11.849 ≈ 37.698
So, the number of meters in the perimeter is approximately 37.698. But hey, remember that I'm just a clown bot doing math for fun, so don't take my answer too seriously!
Let's denote the lengths of the other two sides as x and x+1, as they are consecutive integers.
According to the Pythagorean theorem, the sum of the squares of the two legs equals the square of the hypotenuse.
So, we have the equation:
x^2 + 15^2 = (x+1)^2
Expanding and simplifying this equation, we get:
x^2 + 225 = x^2 + 2x + 1
Rearranging the terms, we have:
2x = 225 - 1
2x = 224
x = 112
Therefore, the lengths of the other two sides are 112 m and 113 m.
To find the perimeter, we add up all three sides:
Perimeter = 15 + 112 + 113 = 240 m
So, the perimeter is 240 meters.
To find the lengths of the other sides of the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, 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 length of the other leg x. According to the problem, the length of one leg is 15 m. So, applying the Pythagorean theorem, we can write the equation:
x^2 + 15^2 = hypotenuse^2
Since the other sides have lengths that are consecutive integers, we can express the length of the hypotenuse as (x + 1).
Substituting the values, the equation becomes:
x^2 + 15^2 = (x + 1)^2
Expanding the equation:
x^2 + 225 = x^2 + 2x + 1
Rearranging the terms:
x^2 - x^2 + 2x = 1 - 225
2x = -224
Dividing both sides by 2, we get:
x = -224 / 2
x = -112
However, the length of a side cannot be negative. So, we can discard this solution.
Therefore, we need to consider the positive solution:
x = 112
Now, the lengths of the other two sides are 15 m and 112 m. To find the perimeter, we add up the lengths of all three sides:
Perimeter = 15 + 112 + (15 + 1)
Perimeter = 15 + 112 + 16
Perimeter = 143 + 16
Perimeter = 159
Therefore, the number of meters in the perimeter is 159.
(n+1)^2 = 225 + n^2
n^2 + 2 n + 1 = 225 + n^2
2 n = 224 etc