In a circle if PQ is the diameter of the circle and R is on the cumference of the circle such that `/_PQR=30^@`, then `/_RPQ = ? `

Options

A. `90^@`
B. `60^@`
C. `30^@`
D. `45^@`

Answer

To solve the problem, we need to find the angle \( \angle RPQ \) given that \( \angle PQR = 30^\circ \) and \( PQ \) is the diameter of the circle.

### Step-by-step Solution:

1. **Understand the Geometry**:
Since \( PQ \) is the diameter of the circle, the angle \( \angle PQR \) is inscribed in a semicircle. According to the properties of circles, any angle inscribed in a semicircle is a right angle (90 degrees).

2. **Identify the Angles**:
We know that:
\[ \angle PQR = 30^\circ \]
We need to find \( \angle RPQ \).

3. **Use the Triangle Sum Property**:
In triangle \( PQR \), the sum of the angles is always 180 degrees. Therefore, we can write:
\[ \angle RPQ + \angle PQR + \angle RQP = 180^\circ \]

4. **Substituting Known Values**:
Since \( \angle PQR = 30^\circ \) and \( \angle RQP = 90^\circ \) (because it is inscribed in a semicircle), we can substitute these values into the equation:
\[ \angle RPQ + 30^\circ + 90^\circ = 180^\circ \]

5. **Simplifying the Equation**:
Combine the known angles:
\[ \angle RPQ + 120^\circ = 180^\circ \]

6. **Isolating \( \angle RPQ \)**:
Now, subtract \( 120^\circ \) from both sides:
\[ \angle RPQ = 180^\circ - 120^\circ \]
\[ \angle RPQ = 60^\circ \]

### Final Answer:
Thus, the measure of \( \angle RPQ \) is \( 60^\circ \).

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