### 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?

**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√25 2 - Now, in the right angle triangle ΔALO

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

⇒ OL^{2}= (

)√25 2 ^{2}+ (

)√25 2 ^{2}

⇒ OL^{2}=

+25 4 25 4

⇒ OL^{2}=50 4

⇒ OL =√50 √4

⇒ OL =

cm√50 2

Now, the side of square LMNO is

cm√50 2

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

Therefore, LT = LQ =

cm√50 4 - Similarly, in the right angle triangle ΔLQT,

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

⇒ QT^{2}= (

)√50 4 ^{2}+ (

)√50 4 ^{2}

⇒ QT^{2}= (

) + (50 16

)50 16

⇒ QT^{2}= (

)100 16

⇒ QT^{2}= (

)25 4

⇒ QT =

cm5 2 - Thus, the perimeter of the square QRST = 4 × QT

= 4 ×5 2

=**10 cm**