### Consider a square ABCD of area 25 cm2. 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?

10

Step by Step Explanation:
1. According to the question, area of the square ABCD = 25 cm2
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: 2. Let us assume a as the side of the square ABCD. Since, the square ABCD has the area 25 cm2.Therefore, we can say that a2 = 25
a = √25 cm2
⇒ AB = a = √25 cm
Since, L and O are the midpoints of AB and AD, respectively, therefore AL = AO =
 √25 2
cm
3. Now, in the right angle triangle ΔALO
OL2 = AL2 + AO2
⇒ OL2 = (
 √25 2
)2 + (
 √25 2
)2
⇒ OL2 =
 25 4
+
 25 4

⇒ OL2 =
 50 4

⇒ OL =
 √50 √4

⇒ OL =
 √50 2
cm
Now, the side of square LMNO is
 √50 2
cm
Since, Q and T are the midpoints of LM and LO respectively.
Therefore, LT = LQ =
 √50 4
cm
4. Similarly, in the right angle triangle ΔLQT,
QT2 = LT2 + LQ2
⇒ QT2 = (
 √50 4
)2 + (
 √50 4
)2
⇒ QT2 = (
 50 16
) + (
 50 16
)
⇒ QT2 = (
 100 16
)
⇒ QT2 = (
 25 4
)
⇒ QT =
 5 2
cm
5. Thus, the perimeter of the square QRST = 4 × QT
= 4 ×
 5 2

= 10 cm 