Question

ABCD is a square PQRS is a square formed by joining the mid-points of AB BC CD and AD respectively. EFGH is a square formed by joining the mid-points of PQ, QR, RS and PS respectively. Find the sum of the perimeters of ABCD, PQRS and EFGH if the area of square EFGH is $25c{m}^2$.

Answer 1

Let the side of the outer square ABCD be a cm
$\Rightarrow$ Side of square PQRS $=\frac{BD}{2}=\frac{\sqrt{2a}}{2}=\frac{a}{\sqrt{2}}=PS$
And the side of square EFGH
$=\frac{GE}{\sqrt{2}}=\frac{PS}{\sqrt{2}}=\frac{\left(\frac{a}{\sqrt{2}}\right)}{\sqrt{2}}=\frac{a}{2}$ cm
$\therefore$ Area of square EFGH = $\frac{a}{2}\times \frac{a}{2}=\frac{a^2}{4}c{m}^2$
Given $\frac{a^2}{4}=25\Rightarrow a^2=100\Rightarrow a=10$
$\therefore$ Perimeter of ABCD = 4a = 40 cm
Perimeter of PQRS = $4\frac{a}{\sqrt{2}}=20\sqrt{2}$cm and perimeter of EFGH = 20 cm
$\therefore$ The required perimeter $=(60+20\sqrt{2})$