Answer: Arc PQ measures 24mm.
Step-by-step explanation: If Arc AB measures 16, remembering that length of an arc is given as
Length of Arc = O/360 x 2πr
Where O is the size of the angle subtended by the arc. We should also note that angle C is the opposite of angle 60, therefore angle C equals 60°. Hence we can now express the length of Arc AB properly as,
16 = {60°/360°} x 2πr
16 = (1/6) x 2πr
By cross multiplication we now have
(16 x 6)/2 = πr
48 = πr
48/π = r
The radius of the larger circle has now been identified as 48/π.
If the radius of the small circle is given as ¾ of the larger circle's radius, then radius of small circle would be
Radius = 3/4 x 48/π
Radius = 36/π
Therefore to calculate the length of arc PQ (which by the way is in the smaller circle). Note that the angle subtended by arc PQ equals 180 - 60 which equals 120° {Sum of angles on a straight line equals 180}
PQ = O/360 x 2πr
PQ = (120/360) x 2π x (36/π)
PQ = (1/3) x 2π x (36/π)
PQ = (2π x 36)/(3 x π)
(π cancels out π, numerator and denominator)
PQ = 24
Therefore arc PQ equals 24 mm.
