In this problem, we are asked to find the radius of the quarter-circle. We can solve it by observing the right-angled triangle inscribed in the quarter circle.

Step-by-Step Approach:

1. Identify the Geometry:

   • The figure includes a quarter-circle with a right-angled triangle inscribed.
   • The sides of the triangle are given as 9 cm, 12 cm, and 16 cm (with the 16 cm as the hypotenuse).

2. Relationship Between Triangle and Circle:

   • The hypotenuse (16 cm) lies on the radius of the quarter-circle.
   • Since this triangle is right-angled, the hypotenuse (which is also a radius) extends from the center of the circle to a point on the circumference.

3. Pythagorean Theorem Confirmation:

   • Let's confirm that the triangle is right-angled using the Pythagorean theorem: $9^2 + 12^2 = 16^2$ $81 + 144 = 256 \quad \text{which is true}.$

   Therefore, this is a valid right-angled triangle, and its hypotenuse represents the radius of the circle.

Conclusion:

The radius of the quarter-circle is 16 cm, as the hypotenuse of the triangle is equal to the radius.

Would you like more details or have further questions?

Here are 5 related questions you might find useful:

1. What is the area of the quarter-circle?
2. How would the problem change if the figure were a semicircle?
3. Can we calculate the area of the triangle inside the quarter-circle?
4. What is the perimeter of the triangle inside the quarter-circle?
5. How does the Pythagorean theorem apply to different right triangles?

Tip: For geometry problems involving circles and triangles, always check how the triangle's sides relate to the radius and use the Pythagorean theorem when needed.