Statement-1 : The points a(3,4), B(2,7) ,C(4,4) dn D(3,5) are such that are of them lies inside the triangle formed by other the points
Statement-2 : Centroid of a triangle always lies inside the triangle

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 not a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True

Explanation

To solve the problem, we need to analyze both statements and determine their validity step by step.

### Step 1: Identify the Points
We have four points:
- A(3, 4)
- B(2, 7)
- C(4, 4)
- D(3, 5)

### Step 2: Understand Statement 1
Statement 1 claims that one of the points A, B, C, or D lies inside the triangle formed by the other three points.

### Step 3: Calculate the Centroid of Triangle ABC
To find out if point D lies inside triangle ABC, we first need to calculate the centroid of triangle ABC (formed by points A, B, and C).

The formula for the centroid (G) of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:
\[ G(x, y) = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3} \right) \]

For triangle ABC:
- A(3, 4)
- B(2, 7)
- C(4, 4)

Calculating the x-coordinate of the centroid:
\[ x = \frac{3 + 2 + 4}{3} = \frac{9}{3} = 3 \]

Calculating the y-coordinate of the centroid:
\[ y = \frac{4 + 7 + 4}{3} = \frac{15}{3} = 5 \]

Thus, the centroid G of triangle ABC is:
\[ G(3, 5) \]

### Step 4: Compare Centroid with Point D
Now we compare the coordinates of the centroid G(3, 5) with point D(3, 5). Since both points are the same, point D lies at the centroid of triangle ABC.

### Step 5: Conclusion for Statement 1
Since the centroid of triangle ABC is at point D, it indicates that point D is not strictly inside the triangle but rather at the centroid. However, for the purpose of this problem, we can conclude that one of the points (in this case, D) is at the centroid of the triangle formed by the other three points.

### Step 6: Understand Statement 2
Statement 2 states that the centroid of a triangle always lies inside the triangle. This statement is true for any triangle, as the centroid is always located within the triangle's boundaries.

### Final Conclusion
Both statements are true:
- Statement 1 is true because point D is the centroid of triangle ABC.
- Statement 2 is true as the centroid of any triangle lies inside it.

Thus, the answer is that both statements are correct.