Final answer:
The correct answer is option A. 30√2 yd. The diagonal length of a square playground with a perimeter of 120 yards is 30√2 yards, using the Pythagorean theorem to calculate the hypotenuse of the square that represents the diagonal.
Step-by-step explanation:
To find the length of a diagonal in a square, we use the Pythagorean theorem, which states that the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides (the sides of the square). Since the perimeter of the square is 120 yards, each side is 120 yards ÷ 4, which equals 30 yards. Therefore, the diagonal can be calculated as the square root of the sum of the squares of the sides: √(30^2 + 30^2), which simplifies to √(900 + 900) and further to √(1800). The square root of 1800 can be expressed as 30√2, so the length of the diagonal is 30√2 yards.
The perimeter of a square is the sum of all its sides. Since the square playground has a perimeter of 120 yards, each side of the square is 120/4 = 30 yards. To find the length of the diagonal, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. In this case, the two sides of the square are equal to 30 yards. Thus, the length of the diagonal can be found by calculating the square root of 30^2 + 30^2 = 2 * 30^2 = 2 * 900 = 1800. Taking the square root of 1800 gives us an approximate value of 42.4 yards. Therefore, the length of the diagonal of the playground is approximately 42.4 yards, which is equivalent to option A: 30√2 yards.