The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is Option1 a rhombus Option2 a rectangle Option3 a square Option4 any parallelogram

Options

A. a rhombus
B. a rectangle
C. a square
D. any parallelogram

Explanation

To solve the problem of determining the figure obtained by joining the midpoints of the sides of a rhombus, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus is a type of quadrilateral where all four sides are of equal length. Additionally, the diagonals of a rhombus bisect each other at right angles.

**Hint:** Remember the properties of a rhombus, especially that all sides are equal and the diagonals bisect each other.

### Step 2: Identify the midpoints of the sides
Let’s label the vertices of the rhombus as A, B, C, and D. The midpoints of the sides AB, BC, CD, and DA can be denoted as P, Q, R, and S respectively.

**Hint:** When finding midpoints, use the formula for the midpoint of a line segment, which is the average of the coordinates of the endpoints.

### Step 3: Apply the Midpoint Theorem
According to the Midpoint Theorem, if you connect the midpoints of two sides of a triangle, the line segment connecting these midpoints is parallel to the third side and half its length.

- For triangle ABC:
- P is the midpoint of AB.
- Q is the midpoint of BC.
- Therefore, PQ is parallel to AC and PQ = 1/2 * AC.

- For triangle ADC:
- S is the midpoint of DA.
- R is the midpoint of CD.
- Therefore, SR is parallel to AC and SR = 1/2 * AC.

**Hint:** Use the Midpoint Theorem to establish relationships between the segments formed by the midpoints and the sides of the rhombus.

### Step 4: Establish relationships between the segments
From the above, we can conclude:
- PQ is parallel to SR and both are equal in length because they are both half of AC.
- Similarly, we can show that PS is parallel to QR and PS = QR.

**Hint:** Remember that in a parallelogram, opposite sides are equal and parallel.

### Step 5: Show that the figure is a parallelogram
Since we have established that both pairs of opposite sides (PQ and SR, PS and QR) are equal and parallel, we can conclude that PQRS forms a parallelogram.

**Hint:** To confirm it’s a parallelogram, verify that both pairs of opposite sides are equal and parallel.

### Step 6: Determine the type of parallelogram
In a rhombus, the diagonals bisect each other at right angles. Since the diagonals of the rhombus are equal, the figure formed (PQRS) must have equal diagonals as well. A parallelogram with equal diagonals is a rectangle.

**Hint:** Recall that a rectangle is defined as a parallelogram with equal diagonals.

### Conclusion
Thus, the figure obtained by joining the midpoints of the sides of a rhombus is a rectangle.

**Final Answer:** Option 2: a rectangle.