Two complementary angles are in the ratio `7: 11`. Find the angles

Options

A. `35^(@) and 75^(@)`
B. `75^(@) and 45^(@)`
C. `35^(@) and 55^(@)`
D. `35^(@) and 36^(@)`

Answer

To solve the problem of finding two complementary angles that are in the ratio of 7:11, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Complementary Angles**:
Complementary angles are two angles whose sum is 90 degrees.

2. **Setting Up the Variables**:
Let the first angle be \( x \) degrees. Since the angles are complementary, the second angle will be \( 90 - x \) degrees.

3. **Using the Ratio**:
According to the problem, the ratio of the two angles is given as \( 7:11 \). This can be expressed as:
\[ \frac{x}{90 - x} = \frac{7}{11} \]

4. **Cross Multiplying**:
To eliminate the fraction, we can cross-multiply:
\[ 11x = 7(90 - x) \]

5. **Expanding the Equation**:
Now, distribute the 7 on the right side:
\[ 11x = 630 - 7x \]

6. **Combining Like Terms**:
Add \( 7x \) to both sides to combine the \( x \) terms:
\[ 11x + 7x = 630 \]
\[ 18x = 630 \]

7. **Solving for \( x \)**:
Divide both sides by 18 to find \( x \):
\[ x = \frac{630}{18} \]
Simplifying this gives:
\[ x = 35 \]

8. **Finding the Second Angle**:
Now, substitute \( x \) back to find the second angle:
\[ 90 - x = 90 - 35 = 55 \]

9. **Conclusion**:
The two angles are \( 35 \) degrees and \( 55 \) degrees.

### Final Answer:
The two complementary angles are \( 35^\circ \) and \( 55^\circ \).

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