Let the circle be with center O and radius 10 cm. Let there be a chord AB
Draw a perpendicular from O on AB to meet at P. The perpendicular from the center divides the chord in two halves. Given, OP=6cm
Thus, In △OAP, Using Pythagoras theorem
OA2=AP2+OP2
102=AP2+62
AP2=64
AP=8 cm
Thus, AB=2AP=16 cm
Now, new chord CD is drawn with length 8 cm. Draw a perpendicular on CD from O to cut CD at N.
Now, In △ONC
OC2=NC2+ON2
102=42+ON2
ON=√84 cm