The area of a square ABCD is `36cm^(2)`. Find the area of the square obtained by joining the midpoints of the sides of the square ABCD.

Options

A. 17 `cm^2`
B. 18 `cm^2`
C. 28 `cm^2`
D. 10 `cm^2`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Find the side length of square ABCD.
Given that the area of square ABCD is \(36 \, \text{cm}^2\), we can use the formula for the area of a square, which is:

\[ \text{Area} = \text{side}^2 \]

Let the side length of the square be \(a\). Then we have:

\[ a^2 = 36 \]

To find \(a\), we take the square root of both sides:

\[ a = \sqrt{36} = 6 \, \text{cm} \]

### Step 2: Determine the midpoints of the sides of square ABCD.
The midpoints of the sides of square ABCD will be:

- Midpoint of side AB: \(P\)
- Midpoint of side BC: \(Q\)
- Midpoint of side CD: \(R\)
- Midpoint of side DA: \(S\)

### Step 3: Calculate the length of the sides of the new square PQRS.
The distance between the midpoints \(P\) and \(Q\) (which forms one side of the new square) can be calculated as follows:

Since \(P\) and \(Q\) are midpoints of sides of length \(6 \, \text{cm}\), the distance \(PQ\) is half the length of the side of square ABCD:

\[ PQ = \frac{1}{2} \times 6 = 3 \, \text{cm} \]

### Step 4: Find the area of square PQRS.
Now, we can find the area of square PQRS using the formula for the area of a square:

\[ \text{Area of PQRS} = \text{side}^2 \]

The side length of square PQRS is \(PQ\), which we found to be \(3 \, \text{cm}\):

\[ \text{Area of PQRS} = 3^2 = 9 \, \text{cm}^2 \]

### Final Answer:
The area of the square obtained by joining the midpoints of the sides of square ABCD is \(9 \, \text{cm}^2\).

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