One leg of a right triangle has a length of 5 m. The other sides have lengths that are consecutive integers. Find these lengths.
The other leg is ____ m
The length of the hypotenuse is ____ m
Let's call one leg of the right triangle "x" and the other leg "x + 1" (since the lengths are consecutive integers).
According to 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, we have:
x^2 + (x + 1)^2 = 5^2
Expanding the equation, we get:
x^2 + (x^2 + 2x + 1) = 25
Combining like terms, we have:
2x^2 + 2x + 1 = 25
Subtracting 25 from both sides, we get:
2x^2 + 2x - 24 = 0
Dividing both sides by 2, we get:
x^2 + x - 12 = 0
Factoring the quadratic equation, we get:
(x + 4)(x - 3) = 0
Setting each factor equal to zero and solving for x, we have:
x + 4 = 0 or x - 3 = 0
If x + 4 = 0, then x = -4. Since the length cannot be negative, we disregard this solution.
If x - 3 = 0, then x = 3.
Therefore, the other leg of the right triangle is 3 m.
To find the length of the hypotenuse, we can substitute the values of the legs into the Pythagorean theorem:
Hypotenuse^2 = x^2 + (x + 1)^2
Hypotenuse^2 = 3^2 + (3 + 1)^2
Hypotenuse^2 = 9 + 16
Hypotenuse^2 = 25
Taking the square root of both sides, we have:
Hypotenuse = √25
Therefore, the length of the hypotenuse is 5 m.
To solve this problem, 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 lengths of the other two sides.
Let's assume the lengths of the other two sides are x and (x + 1) respectively. According to the problem, one leg has a length of 5 m. So, we have:
5^2 + x^2 = (x + 1)^2
Simplifying the equation, we get:
25 + x^2 = x^2 + 2x + 1
Subtracting x^2 from both sides, we have:
25 = 2x + 1
Subtracting 1 from both sides, we get:
24 = 2x
Dividing by 2, we have:
12 = x
So, the other leg has a length of 12 m.
To find the length of the hypotenuse, we can substitute the value of x back into the equation:
c^2 = 5^2 + 12^2
c^2 = 25 + 144
c^2 = 169
Taking the square root of both sides, we get:
c = √169
c = 13
Therefore, the length of the hypotenuse is 13 m.