# The chord of the circle is 6√2 dm and contracts the arc 90 °. Find the circumference and the length of the arc.

From point O, the center of the circle, draw segments OA and OB equal to the radius of the circle.

The central angle AOB is equal to the degree measure of the arc that contracts the chord AB. Angle AOB = 90.

The triangle AOB is isosceles, since OA = OB = R and rectangular, since the angle AOB = 90.

Then, by the Pythagorean theorem, AB ^ 2 = OA ^ 2 + OB ^ 2 = 2 * OA ^ 2.

(6 * √2) ^ 2 = 2 * OA ^ 2.

OA ^ 2 = 36.

ОА = ОВ = R = 6 cm.

Let us determine the length of the arc AB by the formula: L = n * R * α / 180, where R is the radius of the circle, and α is the angle between the radii.

L = n * 6 * 90/180 = 3 * n cm.

Determine the length of the circle.

Lokr = 2 * n * R = 2 * n * 6 = 12 * n cm.

Answer: The length of the arc is 3 * n cm, the circumference is 12 * n cm.