If `D ,Ga n dR` denote respectively the number of degrees, grades and radians in an angle, then

Options

A. `D/(100)=G/(90)=(2R)/pi`
B. `D/(90)=G/(100)=R/pi`
C. `D/(90)=G/(100)=(2R)/pi`
D. `D/(90)=G/(100)=R/(2pi)`

Explanation

To find the relationship between the number of degrees (D), grades (G), and radians (R) in an angle, we can derive equations based on the definitions of these units in terms of right angles.

### Step-by-Step Solution:

1. **Understanding Right Angles**:
- We know that 1 right angle is equivalent to 90 degrees.
- Therefore, we can express degrees in terms of right angles:
\[ D \text{ degrees} = \frac{D}{90} \text{ right angles} \]
- This gives us our first equation:
\[ \theta = \frac{D}{90} \text{ right angles} \quad \text{(Equation 1)} \]

2. **Converting Grades to Right Angles**:
- We also know that 1 right angle is equivalent to 100 grades.
- Thus, we can express grades in terms of right angles:
\[ G \text{ grades} = \frac{G}{100} \text{ right angles} \]
- This gives us our second equation:
\[ \theta = \frac{G}{100} \text{ right angles} \quad \text{(Equation 2)} \]

3. **Converting Radians to Right Angles**:
- We know that 2π radians is equivalent to 2 right angles.
- Therefore, we can express radians in terms of right angles:
\[ R \text{ radians} = \frac{2R}{\pi} \text{ right angles} \]
- This gives us our third equation:
\[ \theta = \frac{2R}{\pi} \text{ right angles} \quad \text{(Equation 3)} \]

4. **Equating the Three Equations**:
- From Equations 1, 2, and 3, we can set them equal to each other:
\[ \frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi} \]

### Final Relationship:
Thus, the relationship between degrees (D), grades (G), and radians (R) can be summarized as:
\[ \frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi} \]