A square is inscribed in a circle of radius r. Express the length of the side of the square in terms of r.
To find the length of the side of the square in terms of the radius, we can use the relationship between the diagonal of the square and the radius of the circle.
In a square, the diagonal is equal to the side length multiplied by the square root of 2 (diagonal = side * sqrt(2)).
We can also see that the diameter of the circle is equal to twice the radius (diameter = 2 * radius).
Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.
Therefore, we can equate the diagonal of the square to the diameter of the circle:
side * sqrt(2) = 2 * radius
To express the length of the side of the square in terms of r, we isolate the side:
side = (2 * radius) / sqrt(2)
Simplifying further, we can rationalize the denominator by multiplying both the numerator and denominator by sqrt(2):
side = (2 * radius * sqrt(2)) / (sqrt(2) * sqrt(2))
= (2 * radius * sqrt(2)) / 2
= radius * sqrt(2)
Therefore, the length of the side of the square, in terms of the radius, is r * sqrt(2).