ABCD is a quadrilateral in which a circle is inscribed.
Statement:1 The length of the sides of the quadrilateral can be A.P.
and Statement: 2: The length of tangents from an external point to a circle are equal.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-15
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-15
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze the two statements regarding the quadrilateral ABCD with an inscribed circle.

### Step 1: Understanding Statement 2
**Statement 2:** The length of tangents from an external point to a circle are equal.

- Let’s consider a circle with center O and an external point P.
- From point P, draw two tangents to the circle, touching the circle at points A and B.
- By the property of tangents to a circle, the lengths of the tangents PA and PB are equal.

**Proof:**
1. Join the center O to points A and B (the points of tangency).
2. The angles ∠OAP and ∠OBP are both 90 degrees (since the radius is perpendicular to the tangent at the point of tangency).
3. In triangles OAP and OBP:
- OA = OB (radii of the same circle)
- OP is common to both triangles.
- ∠OAP = ∠OBP = 90 degrees.
4. By the Side-Angle-Side (SAS) congruence criterion, triangles OAP and OBP are congruent.
5. Therefore, PA = PB.

Thus, **Statement 2 is true.**

### Step 2: Understanding Statement 1
**Statement 1:** The lengths of the sides of the quadrilateral can be in Arithmetic Progression (A.P.).

- For a quadrilateral ABCD with an inscribed circle, the lengths of the sides can be denoted as a, b, c, and d.
- According to the property of tangents from an external point, we can denote the lengths of the tangents from points A, B, C, and D to the points where the circle touches the sides as follows:
- Let the tangents from point A be x, from point B be y, from point C be z, and from point D be w.

- The lengths of the sides of the quadrilateral can be expressed as:
- AB = x + y
- BC = y + z
- CD = z + w
- DA = w + x

- To check if these can be in A.P., we can set:
- a = x + y
- b = y + z
- c = z + w
- d = w + x

- For these to be in A.P., the condition \(2b = a + c\) and \(2c = b + d\) must hold.

- If we assume that the lengths of the tangents (x, y, z, w) can be chosen such that they form an arithmetic sequence, then we can derive that the sides of the quadrilateral can also be in A.P.

Thus, **Statement 1 is also true.**

### Conclusion
Both statements are true, and they are related to each other. The lengths of the sides of a quadrilateral with an inscribed circle can indeed be in A.P., and the lengths of tangents from an external point to the circle are equal.

### Final Answer
Both statements are true, and they are related to each other.

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