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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?
Answer:
10
- 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: - 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
β a = β25 cm2
β AB = a = β25 cm
Since, L and O are the midpoints of AB and AD, respectively, therefore AL = AO =
cmβ25 2 - Now, in the right angle triangle ΞALO
OL2 = AL2 + AO2
β OL2 = (
)2 + (β25 2
)2β25 2
β OL2 =
+25 4 25 4
β OL2 =50 4
β OL =β50 β4
β OL =
cmβ50 2
Now, the side of square LMNO is
cmβ50 2
Since, Q and T are the midpoints of LM and LO respectively.
Therefore, LT = LQ =
cmβ50 4 - Similarly, in the right angle triangle ΞLQT,
QT2 = LT2 + LQ2
β QT2 = (
)2 + (β50 4
)2β50 4
β QT2 = (
) + (50 16
)50 16
β QT2 = (
)100 16
β QT2 = (
)25 4
β QT =
cm5 2 - Thus, the perimeter of the square QRST = 4 Γ QT
= 4 Γ5 2
= 10 cm
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