Two tangents AB and AC are drawn from an external point A to a circle with centre O . If they are inclined to each other at an angle of `100^@` then what is the value of `angleBOC.`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the configuration
We have a circle with center O and two tangents AB and AC drawn from an external point A. The angle between the tangents AB and AC is given as 100°.

### Step 2: Identify the angles
Since AB and AC are tangents to the circle, the radius at the point of tangency (let's say points B and C) is perpendicular to the tangent. Therefore, we have:
- Angle OAB = 90°
- Angle OAC = 90°

### Step 3: Use the properties of triangles
Since the tangents AB and AC are equal in length (tangents drawn from an external point to a circle are equal), triangles OAB and OAC are congruent. Thus:
- OB = OC (radii of the circle)
- OA is common to both triangles

### Step 4: Establish relationships between angles
From the congruence of triangles OAB and OAC, we can say:
- Angle OAB = Angle OAC

Let’s denote Angle OAB as x. Then:
- Angle OAB + Angle OAC + Angle BAC = 100°
- 2x + 100° = 100°
- 2x = 100° - 100°
- 2x = 0°
- x = 50°

### Step 5: Find angle BOA
Now, we know:
- Angle OAB = 50°
- Angle OBA = 90° (since it is a right angle)

Using the triangle sum property in triangle OAB:
- Angle OAB + Angle OBA + Angle BOA = 180°
- 50° + 90° + Angle BOA = 180°
- Angle BOA = 180° - 140°
- Angle BOA = 40°

### Step 6: Find angle BOC
Since triangles OAB and OAC are congruent, Angle COA is also 40°.
Thus, we can find Angle BOC:
- Angle BOC = Angle BOA + Angle COA
- Angle BOC = 40° + 40°
- Angle BOC = 80°

### Final Answer
The value of angle BOC is 80°.

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