Explanation:
To understand why the figure formed by joining the mid-points of the adjacent sides of a square is a square, let's consider a square ABCD.

Definition:
- A square is a quadrilateral with all four sides equal in length and all four angles equal to 90 degrees.

Construction:
- Let P, Q, R, and S be the midpoints of AB, BC, CD, and DA respectively.
- Join PQ, QR, RS, and SP.

Proof:
- To prove that the figure formed by joining the mid-points of the adjacent sides of a square is a square, we need to show that all four sides are equal in length and all four angles are equal to 90 degrees.

All four sides are equal in length:
- In a square ABCD, all four sides are equal. Therefore, AB = BC = CD = DA.
- In the figure formed by joining the mid-points of the adjacent sides of the square, we have PQ || AB, QR || BC, RS || CD, and SP || DA.
- By the midpoint theorem, we know that PQ = QR = RS = SP.
- Therefore, all four sides of the figure formed by joining the mid-points are equal in length.

All four angles are equal to 90 degrees:
- In a square ABCD, all four angles are equal to 90 degrees.
- In the figure formed by joining the mid-points of the adjacent sides of the square, we can see that PQ is parallel to AB and QR is parallel to BC.
- Since AB and BC are perpendicular to each other, PQ and QR are also perpendicular to each other.
- Similarly, RS is perpendicular to CD and SP is perpendicular to DA.
- Therefore, all four angles of the figure formed by joining the mid-points are equal to 90 degrees.

Conclusion:
- Since all four sides of the figure formed by joining the mid-points of the adjacent sides of a square are equal in length and all four angles are equal to 90 degrees, the figure is a square.
- Hence, option 'D', square, is the correct answer.