Smaller square

Let the points A, B, C and D are midpoints of the sides of the square PQRS. If the area of PQRS is 100 sq cm, what is the area of the smaller square ABCD?

Correct answer:

S₂ =  50 cm²

Step-by-step explanation:

$$\begin{gathered} S_1=100{\text{cm}}^2 \\ {S}_1=a^2 \\ a=\sqrt{S_1}=\sqrt{100}=10\text{cm} \\ (a/2{)}^2+(a/2{)}^2=b^2 \\ b=\sqrt{2\cdot (a/2{)}^2}=\sqrt{2\cdot (10/2{)}^2}=5\sqrt{2}\text{cm}\doteq 7.0711\text{cm} \\ {S}_2=b^2=7.0711^2=50{\text{cm}}^2 \end{gathered}$$

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