I is the incentre of a triangle PQR and `angle = 107^@` then find the value `angle P`.

Options

A. 56°
B. 34°
C. 26°
D. 42°

Answer

To solve the problem, we need to find the value of angle P in triangle PQR, given that I is the incentre of the triangle and angle I is 107 degrees.

### Step-by-Step Solution:

1. **Understand the Incentre**: The incentre (I) of a triangle is the point where the angle bisectors of the triangle intersect. It is also the center of the circle that can be inscribed within the triangle.

2. **Use the Incentre Angle Formula**: The relationship between the angle at the incentre (I) and the angles of the triangle can be expressed as:
\[ \angle I = 90^\circ + \frac{\angle P}{2} \]
where \( \angle P \) is the angle at vertex P of triangle PQR.

3. **Substitute the Given Value**: We know that \( \angle I = 107^\circ \). Therefore, we can set up the equation:
\[ 107 = 90 + \frac{\angle P}{2} \]

4. **Isolate \( \angle P \)**: To find \( \angle P \), we first subtract 90 from both sides:
\[ 107 - 90 = \frac{\angle P}{2} \]
This simplifies to:
\[ 17 = \frac{\angle P}{2} \]

5. **Multiply by 2**: To solve for \( \angle P \), multiply both sides by 2:
\[ \angle P = 17 \times 2 \]
Thus,
\[ \angle P = 34^\circ \]

### Final Answer:
The value of angle P is \( 34^\circ \).

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