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
Is my answer correct?
The other leg is 12 m
The hypotenuse is 13 m
Correct
The other sides :
x
and
x + 1
where x + 1 is the hypotenuse
So Pythagorean theorem:
( x + 1 ) ^ 2 = x ^ 2 + 5 ^ 2
x ^ 2 + 2 x + 1 = x ^ 2 + 25 Both x ^ 2 get cancelled
2 x + 1 = 25
2 x = 25 - 1 = 25 - 1 = 24
x = 24 / 2 = 12 m
hypotenuse
x + 1 = 12 + 1 = 13 m
It is 12
Hypo is 13
Well, I must say, your answer is quite right on the money! You've hit the nail on the head with those measurements. It seems like you've got a good grasp of those Pythagorean triplets! Keep up the spiffy work!
To find the lengths of the other leg and the hypotenuse of the right triangle, we can use the Pythagorean theorem. The Pythagorean theorem 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 denote the length of the other leg as x, where x is an integer.
According to the given information, one leg of the right triangle has a length of 5 m. Therefore, we have:
(5)^2 + (x)^2 = (hypotenuse)^2
Simplifying the equation:
25 + x^2 = hypotenuse^2
Since the lengths of the other two sides are consecutive integers, we can write x as x and (x+1) representing the consecutive integers.
Now, we substitute the values into the equation:
25 + x^2 = (x+1)^2
Expanding the equation:
25 + x^2 = x^2 + 2x + 1
Simplifying further:
25 = 2x + 1
Subtracting 1 from both sides:
24 = 2x
Dividing both sides by 2:
12 = x
Therefore, the length of the other leg is 12 m.
To find the length of the hypotenuse, we substitute the value of x into the equation:
hypotenuse = x + 1
hypotenuse = 12 + 1
hypotenuse = 13 m
So, your answer is correct. The other leg of the right triangle is 12 m, and the length of the hypotenuse is 13 m.