The chord of the circle is 5 roots of 2 and subtracts the arc by 90 degrees. Find the arc length and area of the corresponding sector

From the center O of the circle, we construct the radii OA and OB to the ends of the chord.

By condition, the arc AB contracts the arc 90, then the central angle AOB is equal to 90, and therefore the triangle AOB is rectangular, and since OA = OB = R, it is isosceles, then the angle OAB = OBA = 45.

Determine the radius of the circle from the triangle AOB.

Cos45 = R / AB.

R = AB * Cos45 = 5 * √2 * √2 / 2 = 5 cm.

Let us determine the length of the smaller chord AB.

Lav = n * R * 90/180 = n * 5 * 90/180 = 2.5 * p.

Let’s define the area of the OAB sector.

Ssec = n * R2 * 90/360 = n * 50 * 90/360 = 12.5 * n cm2.

Answer: The length of the arc is 2.5 * n, the area of the sector is 12.5 * n cm2.