Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.

Answer

To prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus, we can follow these steps:

### Step-by-Step Solution:

1. **Consider a Rectangle**: Let the rectangle be \(ABCD\) where \(A\) and \(B\) are the top vertices, and \(C\) and \(D\) are the bottom vertices.

2. **Identify Mid-Points**: Let \(P\), \(Q\), \(R\), and \(S\) be the mid-points of sides \(AB\), \(BC\), \(CD\), and \(DA\) respectively.

3. **Use the Mid-Point Theorem**: According to the mid-point theorem, the line segment joining two mid-points of a triangle is parallel to the third side and half its length.

4. **Calculate Lengths**:
- In triangle \(ABD\), \(P\) and \(S\) are mid-points of sides \(AB\) and \(AD\) respectively. Therefore, \(PS\) is parallel to \(BD\) and \(PS = \frac{1}{2} BD\).
- In triangle \(BCD\), \(Q\) and \(R\) are mid-points of sides \(BC\) and \(CD\) respectively. Therefore, \(QR\) is parallel to \(BD\) and \(QR = \frac{1}{2} BD\).

5. **Establish Equality of Sides**: Since both \(PS\) and \(QR\) are equal to \(\frac{1}{2} BD\), we have:
\[ PS = QR \]

6. **Repeat for Other Sides**:
- In triangle \(ABC\), \(P\) and \(Q\) are mid-points of sides \(AB\) and \(BC\) respectively. Therefore, \(PQ\) is parallel to \(AC\) and \(PQ = \frac{1}{2} AC\).
- In triangle \(ADC\), \(R\) and \(S\) are mid-points of sides \(CD\) and \(DA\) respectively. Therefore, \(RS\) is parallel to \(AC\) and \(RS = \frac{1}{2} AC\).

7. **Establish Equality of Other Sides**: Since both \(PQ\) and \(RS\) are equal to \(\frac{1}{2} AC\), we have:
\[ PQ = RS \]

8. **Conclude that All Sides are Equal**: We have established that:
\[ PS = QR \quad \text{and} \quad PQ = RS \]
Since \(AC = BD\) (as diagonals of a rectangle are equal), we conclude:
\[ PS = QR = PQ = RS \]

9. **Final Conclusion**: Since all four sides \(PQ\), \(QR\), \(RS\), and \(PS\) are equal, the figure \(PQRS\) is a rhombus.