The figure obtained by joining the mid-points of the adjacent sides of a rectangle of sides 8 cm and 6 cm, is

Options

A. a rectangle of area `24 cm^(2)`
B. a square of area `25 cm^(2)`
C. a trapezium of area `24 cm^(2)`
D. a rhombus of area `24 cm^(2)`

Explanation

To solve the problem, we need to find the figure obtained by joining the midpoints of the adjacent sides of a rectangle with sides 8 cm and 6 cm, and then calculate the area of that figure.

### Step-by-Step Solution:

1. **Identify the Rectangle**:
- We have a rectangle ABCD where AB = 8 cm (length) and BC = 6 cm (breadth).
- Let's label the vertices: A(0, 0), B(8, 0), C(8, 6), D(0, 6).

**Hint**: Start by sketching the rectangle and labeling the vertices.

2. **Find the Midpoints**:
- The midpoints of the sides are:
- E (midpoint of AB) = ((0 + 8)/2, (0 + 0)/2) = (4, 0)
- F (midpoint of BC) = ((8 + 8)/2, (0 + 6)/2) = (8, 3)
- G (midpoint of CD) = ((8 + 0)/2, (6 + 6)/2) = (4, 6)
- H (midpoint of DA) = ((0 + 0)/2, (6 + 0)/2) = (0, 3)

**Hint**: Use the midpoint formula: Midpoint = ((x1 + x2)/2, (y1 + y2)/2).

3. **Connect the Midpoints**:
- Now, connect the points E, F, G, and H in order to form a quadrilateral.

**Hint**: Draw lines connecting the points E, F, G, and H sequentially.

4. **Identify the Figure**:
- The quadrilateral EFGH formed by joining the midpoints is a rhombus. This is because the diagonals of the rhombus bisect each other at right angles and the opposite sides are equal.

**Hint**: Recall that in a rectangle, joining the midpoints of adjacent sides always forms a rhombus.

5. **Calculate the Diagonals**:
- The diagonals of the rhombus are:
- Diagonal 1 (EF) = Distance between E(4, 0) and G(4, 6) = 6 cm
- Diagonal 2 (FH) = Distance between F(8, 3) and H(0, 3) = 8 cm

**Hint**: Use the distance formula to find the lengths of the diagonals.

6. **Calculate the Area of the Rhombus**:
- The area of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2} \]
- Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 6 \times 8 = \frac{48}{2} = 24 \text{ cm}^2 \]

**Hint**: Remember to multiply the diagonals and divide by 2 to find the area.

### Final Answer:
The figure obtained by joining the midpoints of the adjacent sides of the rectangle is a rhombus with an area of 24 cm².