### Consider a square ABCD of area 25 cm^{2}. L is the midpoint of AB, M the midpoint of BC, N the midpoint of CD, and O the midpoint of DA. These points are used to construct a new square LMNO. The same process is repeated on LMNO to construct a smaller square QRST (where Q is the midpoint of LM and so on). What is the perimeter of square QRST?

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**Answer: **10

**Step by Step Explanation: **- According to the question, area of the square ABCD = 25 cm
^{2}

Given, L is the midpoint of AB, M the midpoint of BC, N the midpoint of CD, and O the midpoint of DA. These points are used to construct a new square LMNO. The same process is repeated on LMNO to construct a smaller square QRST (where, Q is the midpoint of LM and so on).

The following figure shows the mentioned constructions:

- Let us assume
*a* as the side of the square ABCD. Since, the square ABCD has the area 25 cm^{2}.Therefore, we can say that *a*^{2} = 25

⇒ *a* = √25 cm^{2}

⇒ AB = *a* = √25 cm

Since, L and O are the midpoints of AB and AD, respectively, therefore AL = AO = cm - Now, in the right angle triangle ΔALO

OL^{2} = AL^{2} + AO^{2}

⇒ OL^{2} = ( )^{2} + ( )^{2}

⇒ OL^{2} = +

⇒ OL^{2} =

⇒ OL =

⇒ OL = cm

Now, the side of square LMNO is cm

Since, Q and T are the midpoints of LM and LO respectively.

Therefore, LT = LQ = cm - Similarly, in the right angle triangle ΔLQT,

QT^{2} = LT^{2} + LQ^{2}

⇒ QT^{2} = ( )^{2} + ( )^{2}

⇒ QT^{2} = ( ) + ( )

⇒ QT^{2} = ( )

⇒ QT^{2} = ( )

⇒ QT = cm - Thus, the perimeter of the square QRST = 4 × QT

= 4 ×

= **10 cm**