Question

In the given figure, a circle with centre O is inscribed inside $\Delta LMN$. A and B are the points of tangency. Find $\angle ANB$.

Hint:

Remember that the angle between two tangents drawn from an external point and the angle subtended by the line segments joining the points of contact at the centre are supplementary.
Once you find interior $\angle AOB = 120^\circ$, simply calculate $180^\circ - 120^\circ = 60^\circ$ directly.

Answer 1

Step 1: Understanding the Question:
We are given a circle with centre $O$ inscribed in a triangle $LMN$.
$A$ and $B$ are points of tangency on the sides of the triangle meeting at the vertex $N$.
The reflex angle at the centre is marked as reflex $\angle AOB = 240^\circ$.
We need to find the value of $\angle ANB$.

Step 2: Key Formula or Approach:
1. The sum of the angles around a point is $360^\circ$. This helps us find the interior angle $\angle AOB$:
\[ \text{Interior } \angle AOB = 360^\circ - \text{Reflex } \angle AOB \]
2. Tangents drawn from an external point to a circle are perpendicular to the radius at the point of contact:
\[ \angle OAN = 90^\circ \text{ and } \angle OBN = 90^\circ \]
3. In quadrilateral $OANB$, the sum of all interior angles is $360^\circ$:
\[ \angle OAN + \angle ANB + \angle OBN + \angle AOB = 360^\circ \]
This simplifies to:
\[ \angle ANB + \angle AOB = 180^\circ \]

Step 3: Detailed Explanation:

• 1. Determine the interior angle $\angle AOB$ from the given reflex angle $240^\circ$:
\[ \angle AOB = 360^\circ - 240^\circ = 120^\circ \]

• 2. State the geometric relations between radii and tangents:
- $OA \perp LN \implies \angle OAN = 90^\circ$
- $OB \perp MN \implies \angle OBN = 90^\circ$

• 3. Apply the angle sum property to the quadrilateral $OANB$:
\[ \angle OAN + \angle ANB + \angle OBN + \angle AOB = 360^\circ \]

• 4. Substitute the known values:
\[ 90^\circ + \angle ANB + 90^\circ + 120^\circ = 360^\circ \]

• 5. Simplify the expression:
\[ \angle ANB + 300^\circ = 360^\circ \]

• 6. Solve for $\angle ANB$:
\[ \angle ANB = 360^\circ - 300^\circ = 60^\circ \]


Step 4: Final Answer:
The angle $\angle ANB$ is $60^\circ$.