a right triangle has a hypotenuse that is 59 miles long. One side is 1.05 longer than the other. What are the lengths of the two legs
a^2 + (a+1.05)^2 = 59^2
a^2 + a^2 + 2.1 a + 1.1025 = 3481
2 a^2 + 2.1 a = 3480
a^2 + 1.05 a - 1740 = 0
https://www.wolframalpha.com/input/?i=a^2+%2B+1.05+a+-+1740
a = 41.2
b = 42.2
To find the lengths of the two legs of a 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 lengths of the two other sides.
Let's denote the length of one leg as "x" miles. Since the other leg is 1.05 longer, we can represent its length as "x + 1.05" miles.
According to the Pythagorean Theorem, we can set up the equation as follows:
x^2 + (x + 1.05)^2 = 59^2
Now, let's solve for "x" by simplifying and solving the equation step by step:
Expanding the squared term:
x^2 + (x^2 + 2.1x + 1.1025) = 59^2
Combining like terms:
2x^2 + 2.1x + 1.1025 = 59^2
Moving all terms to one side:
2x^2 + 2.1x + 1.1025 - 59^2 = 0
Calculating the value of 59^2:
2x^2 + 2.1x + 1.1025 - 3481 = 0
Simplifying further:
2x^2 + 2.1x - 3479.8975 = 0
Now, we have a quadratic equation in the form "ax^2 + bx + c = 0," where "a" equals 2, "b" equals 2.1, and "c" equals -3479.8975. We can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the respective values, we have:
x = (-(2.1) ± √((2.1)^2 - 4(2)(-3479.8975))) / (2(2))
Simplifying further:
x = (-2.1 ± √(4.41 + 27759.59)) / 4
Calculating the values under the square root:
x = (-2.1 ± √27763) / 4
Now, we have two possible solutions for "x." Let's calculate them separately:
Solution 1:
x = (-2.1 + √27763) / 4
Calculating the square root:
x = (-2.1 + 166.546) / 4
x ≈ 41.8615
Solution 2:
x = (-2.1 - √27763) / 4
Calculating the square root:
x = (-2.1 - 166.546) / 4
x ≈ -41.7115
Since the length of a side cannot be negative, we discard the second solution.
Therefore, the length of one leg is approximately 41.8615 miles.
Since the other leg is 1.05 miles longer, we can calculate its length as:
x + 1.05 ≈ 41.8615 + 1.05 ≈ 42.9115 miles.
Hence, the lengths of the two legs are approximately 41.8615 miles and 42.9115 miles, respectively.