A chord 8 cm long contracts an arc of 90 degrees, find the length of the diameter segment from

A chord 8 cm long contracts an arc of 90 degrees, find the length of the diameter segment from the center of the circle to the chord.

Let’s draw the radii of the circle OB and OA, then the triangle AOB is isosceles. Since the chord AB contracts the arc at 90, the central angle resting on this chord is also equal to 90. Then the triangle AOB is isosceles and right-angled.

Let us draw from the center of the circle the height OH of the triangle AOB, which is also the median and bisector of the triangle. AH = BH = AB / 2 = 8/2 = 4 cm.

Angle ABH = AOB / 2 = 90/2 = 450.

Let us determine the length of the leg OH in the right-angled triangle AOH.

tgAOH = AH / OH.

OH = AH / tg45 = 4/1 = 4 cm.

Answer: The distance from the center O to the chord is 4 cm.