To solve this problem, we need to find the radius of the circle, given that [tex]\( RS \)[/tex] is the diameter and [tex]\( P \)[/tex] is a point on the circumference. We also know the lengths [tex]\( PR = 12 \, \text{cm} \)[/tex] and [tex]\( PS = 16 \, \text{cm} \)[/tex].
Since [tex]\( RS \)[/tex] is the diameter of the circle and [tex]\( P \)[/tex] is on the circumference, triangle [tex]\( PRS \)[/tex] is a right triangle, according to Thales' theorem.
1. Apply the Pythagorean theorem to triangle [tex]\( PRS \)[/tex]:
[tex]\[ RS^2 = PR^2 + PS^2 \][/tex]
2. Plug in the given lengths [tex]\( PR = 12 \, \text{cm} \)[/tex] and [tex]\( PS = 16 \, \text{cm} \)[/tex]:
[tex]\[ RS^2 = 12^2 + 16^2 \][/tex]
3. Calculate the squares:
[tex]\[ 12^2 = 144 \][/tex]
[tex]\[ 16^2 = 256 \][/tex]
4. Add the squared lengths:
[tex]\[ RS^2 = 144 + 256 = 400 \][/tex]
5. Find [tex]\( RS \)[/tex] by taking the square root of both sides:
[tex]\[ RS = \sqrt{400} = 20 \, \text{cm} \][/tex]
Since [tex]\( RS \)[/tex] is the diameter of the circle, the radius [tex]\( r \)[/tex] is half the diameter:
[tex]\[ r = \frac{RS}{2} = \frac{20}{2} = 10 \, \text{cm} \][/tex]
Therefore, the radius of the circle is [tex]\( 10 \, \text{cm} \)[/tex].
So, the correct choice is:
(ii) 10 cm
