A square has a side length of 95. What is the length of the diagonal of the square? Express your answer in simplest radical form.
Pitagorin teorem:
diagonal = sqrt ( 95 ^ 2 + 95 ^ 2 )
diagonal = sqrt ( 2 * 9025 )
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Remark :
9025 = 5 * 5 * 19 * 19
9025 = 5 ^ 2 * 19 ^ 2
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diagonal = sqrt ( 2 * 9025 )
diagonal = sqrt ( 2 * 5 ^ 2 * 19 ^ 2 )
diagonal = 5 * 19 * sqrt ( 2 )
diagonal = 95 * sqrt ( 2 )
diagonal = sqrt ( 95 ^ 2 + 95 ^ 2 )
diagonal = sqrt ( 2 * 95 ^ 2 )
diagonal = sqrt ( 2 * 9025 )
d = sqrt ( 95 ^ 2 + 95 ^ 2 )
d = sqrt ( 2 95 ^ 2 )
d = sqrt ( 95 ^ 2 ) * sqrt ( 2 )
d = 95 sqrt ( 2 )
To find the length of the diagonal of a square, we can use the Pythagorean Theorem. In a square, the diagonal forms a right triangle with the sides of the square.
The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. In this case, the side length of the square is the leg of the right triangle, and the diagonal is the hypotenuse.
Let's call the side length of the square "s" and the length of the diagonal "d". According to the Pythagorean Theorem, we have:
s^2 + s^2 = d^2
Since the square has equal sides, we can simplify the equation to:
2s^2 = d^2
To find the length of the diagonal, we need to solve for "d". We can start by isolating "d^2" by dividing both sides of the equation by 2:
d^2 = 2s^2
Taking the square root of both sides, we get:
d = √(2s^2)
Now, we can substitute the given side length "95" for "s" in the equation:
d = √(2(95)^2)
Simplifying further, we have:
d = √(2 * 9025)
d = √(18050)
Therefore, the length of the diagonal of the square with a side length of 95 is √18050, which is the simplest radical form.