In cyclic quadrilateral `PQRS`, `7/_ P = 2/_R` Then,`/_P=`

Options

A. `20^(@)`
B. `40^(@)`
C. `140^(@)`
D. `100^(@)`

Answer

To solve the problem, we need to find the value of angle P in the cyclic quadrilateral PQRS, given that \(7 \times \angle P = 2 \times \angle R\).

### Step-by-Step Solution:

1. **Understand the properties of cyclic quadrilaterals**: In a cyclic quadrilateral, the sum of opposite angles is equal to 180 degrees. Therefore, we can write:
\[ \angle P + \angle R = 180^\circ \]

2. **Express angle R in terms of angle P**: From the equation above, we can express angle R as:
\[ \angle R = 180^\circ - \angle P \]

3. **Substitute angle R into the given equation**: We know from the problem statement that:
\[ 7 \times \angle P = 2 \times \angle R \]
Substituting the expression for angle R:
\[ 7 \times \angle P = 2 \times (180^\circ - \angle P) \]

4. **Expand the equation**: Distributing the 2 on the right side gives:
\[ 7 \times \angle P = 360^\circ - 2 \times \angle P \]

5. **Combine like terms**: To combine the terms involving angle P, add \(2 \times \angle P\) to both sides:
\[ 7 \times \angle P + 2 \times \angle P = 360^\circ \]
This simplifies to:
\[ 9 \times \angle P = 360^\circ \]

6. **Solve for angle P**: Divide both sides by 9 to find angle P:
\[ \angle P = \frac{360^\circ}{9} = 40^\circ \]

### Final Answer:
\[ \angle P = 40^\circ \]