PQRS is a square with side 10 cm. A, B, C and D are mid-points of PQ, QR, RS and SP respectively. Then the perimeter of the square ABCD so formed is

Options

A. `10sqrt(2) cm`
B. `20sqrt(2) cm`
C. `25 sqrt(2) cm`
D. `15sqrt(2) cm`

Answer

To find the perimeter of square ABCD formed by the midpoints of square PQRS, we can follow these steps:

### Step 1: Identify the square PQRS
Given that PQRS is a square with each side measuring 10 cm, we can denote the vertices as follows:
- P(0, 0)
- Q(10, 0)
- R(10, 10)
- S(0, 10)

### Step 2: Find the midpoints A, B, C, and D
The midpoints of the sides of the square PQRS are calculated as follows:
- A is the midpoint of PQ:
\[ A = \left(\frac{0 + 10}{2}, \frac{0 + 0}{2}\right) = (5, 0) \]
- B is the midpoint of QR:
\[ B = \left(\frac{10 + 10}{2}, \frac{0 + 10}{2}\right) = (10, 5) \]
- C is the midpoint of RS:
\[ C = \left(\frac{10 + 0}{2}, \frac{10 + 10}{2}\right) = (5, 10) \]
- D is the midpoint of SP:
\[ D = \left(\frac{0 + 0}{2}, \frac{10 + 0}{2}\right) = (0, 5) \]

### Step 3: Calculate the length of side AD
To find the length of side AD, we can use the distance formula:
\[ AD = \sqrt{(5 - 0)^2 + (0 - 5)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \text{ cm} \]

### Step 4: Calculate the perimeter of square ABCD
Since ABCD is a square, all sides are equal. Therefore, the perimeter \(P\) of square ABCD can be calculated as:
\[ P = 4 \times \text{side length} = 4 \times AD = 4 \times 5\sqrt{2} = 20\sqrt{2} \text{ cm} \]

### Final Answer
The perimeter of square ABCD is \(20\sqrt{2} \text{ cm}\).

---