Answer:
the perimeter of the triangle is between 14.32 and 14.33.
Explanation:
To find the perimeter of the triangle, we need to determine the lengths of the legs and then add them together with the hypotenuse.
Given that the ratio of the lengths of the legs is 3 to 1, we can let the lengths be 3x and x, where x is a positive number.
According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Using this, we can set up the equation:
(3x)^2 + x^2 = (√40)^2
Simplifying the equation step by step:
1. Expand and simplify the left side of the equation:
9x^2 + x^2 = 40
2. Combine like terms:
10x^2 = 40
3. Divide both sides of the equation by 10 to solve for x^2:
x^2 = 4
4. Take the square root of both sides to solve for x:
x = √4
x = 2
Now that we know the length of one leg is 2, we can find the length of the other leg by multiplying it by 3:
3x = 3(2) = 6
The lengths of the legs are 2 and 6, and the length of the hypotenuse is √40.
To find the perimeter, we add the lengths of the three sides:
Perimeter = 2 + 6 + √40
To approximate the perimeter, we can calculate the square root of 40:
√40 ≈ 6.32
So, the perimeter of the triangle is between 2 + 6 + 6.32 = 14.32 and 2 + 6 + 6.33 = 14.33.