Choose the CORRECT option about the two tangents drawn to the circle from the external point?
1. The tangents subtend equal angles at the centre of the circle.
2. The tangents drawn from the external point to a circle are equal in length.
3. The tangents are equally inclined to the segment which joins the centre to that point.

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1, 2 and 3
D. 1 and 3 only

Answer

To solve the question regarding the properties of tangents drawn from an external point to a circle, we will analyze each statement step by step.

### Step-by-Step Solution:

**Step 1: Understanding the Geometry of Tangents**
- When two tangents are drawn from an external point to a circle, they touch the circle at distinct points. Let’s denote the external point as P, the center of the circle as O, and the points of tangency as A and B.

**Hint 1:** Visualize the scenario by drawing a circle, marking the center, the external point, and the points where the tangents touch the circle.

**Step 2: Analyzing Statement 1**
- **Statement 1:** "The tangents subtend equal angles at the center of the circle."
- The angles ∠AOP and ∠BOP are formed by the line segments OA and OB (radii) and the line segment OP (from the external point to the center). Since the triangles OAP and OBP are congruent (by the Side-Angle-Side (SAS) criterion), the angles ∠AOP and ∠BOP are equal.

**Hint 2:** Remember that congruent triangles have equal corresponding angles.

**Step 3: Analyzing Statement 2**
- **Statement 2:** "The tangents drawn from the external point to a circle are equal in length."
- The lengths of the tangents PA and PB are equal. This is a well-known property of tangents drawn from an external point to a circle.

**Hint 3:** Use the property that tangents from a common external point to a circle are equal in length.

**Step 4: Analyzing Statement 3**
- **Statement 3:** "The tangents are equally inclined to the segment which joins the center to that point."
- The angles ∠AOP and ∠BOP are equal (as established in Step 2), which means the tangents PA and PB are equally inclined to the line segment OP.

**Hint 4:** Recall that equal angles imply equal inclination to a line.

**Step 5: Conclusion**
- Since all three statements are true based on the properties of tangents and congruent triangles, the correct option is that all statements are correct.

### Final Answer:
All three statements (1, 2, and 3) about the tangents drawn from an external point to a circle are correct.