Final answer:
The radius and diameter of a circle, as well as the diameter and circumference, are directly proportional with constants of 2 and π, respectively. The radius and circumference are also proportional with a constant of 2π. However, the radius and area of a circle are not proportional.
Step-by-step explanation:
To determine whether the given quantities are proportional, we need to understand their relationships:
- a. Radius and diameter of a circle are directly proportional because the diameter is always twice the length of the radius. The constant of proportionality (k) is 2, since Diameter = 2 x Radius.
- b. Radius and circumference of a circle are directly proportional. The circumference (C) of a circle is proportional to its radius (r) by a factor of 2π (C = 2πr). So, the constant of proportionality is 2π.
- c. Radius and area of a circle are not proportional. While both increase as the other does, the area is proportional to the square of the radius (A = πr²), which is a non-linear relationship.
- d. Diameter and circumference of a circle are directly proportional with the constant of proportionality being π, given by the formula C = πd, where d is the diameter.
Understanding these relationships helps in solving geometry problems and grasping the concepts of rate, proportion, and ratios.