Which of the following steps of construction is incorrect while drawing a tangent to a circle of radius 5 cm and making an angle of `30^(@)` with a line passing through the centre ?
Steps of Construction
Step I : Draw a circle with centre O and radius 2.5 cm.
Step II : Draw a radius OA of this circle and produce it to B.
Step III : Construct and angle `angle AOP` equal to the complement of `30^(@)` i.e. equal to `150^(@)`.
Step IV : Draw perpendicular to OP at P which intersects OA produced at Q.
Clearly, PQ is the desired tangent such that `angle OQP = 30^(@)`.

Options

A. Both I and III
B. Only III
C. Both III and IV
D. Only I

Answer

To solve the problem of identifying the incorrect steps in the construction of a tangent to a circle of radius 5 cm that makes an angle of 30 degrees with a line passing through the center, we will analyze each step provided in the question.

### Step-by-Step Solution:

1. **Step I**: Draw a circle with center O and radius 5 cm.
- **Correction**: The original step states to draw a circle with a radius of 2.5 cm, which is incorrect. The radius should be 5 cm as per the problem statement.

2. **Step II**: Draw a radius OA of this circle and produce it to B.
- **Analysis**: This step is correct. Drawing a radius OA and extending it to point B is a valid step in the construction.

3. **Step III**: Construct an angle AOP equal to the complement of 30 degrees, i.e., equal to 150 degrees.
- **Correction**: This step is incorrect. The complement of 30 degrees is not 150 degrees; it is 60 degrees. Therefore, angle AOP should be constructed as 60 degrees, not 150 degrees.

4. **Step IV**: Draw a perpendicular to OP at P which intersects OA produced at Q.
- **Analysis**: This step is correct. Drawing a perpendicular from point P to line OP and extending it to intersect OA at Q is a valid construction step.

5. **Conclusion**: The incorrect steps in the construction are Step I and Step III.

### Final Answer:
- The incorrect steps are Step I (drawing the wrong radius) and Step III (constructing the wrong angle).

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