Four circles of unit radius are drawn touching each other. A circle is drawn in between these four circles, touching these four circles. A square is inscribed in this circle. What is the length of the side of the square.
Answers
Answer:
2- √2 cm
Step-by-step explanation:
Given:- In the figure, four circles of radii(r = 1cm) are touching each other. A circle is drawn between these four circles, touching all four circles. A square GIFH is inscribed in the circle. A, B, C, D are centres of four circles respectively. E is the centre of a smaller circle drawn between these four circles.
Construction:- Join AB, AC, and BC.
Sol:- AB = BC = 2r = 2cm
∠ABC = 90° (By construction)
In ∆ABC,
AB² + BC² = AC²2
2² + 2² = AC²
AC = 2√2 cm
Now, GF = AC - (AG + CF)
= 2√2 - (1 + 1) [AG = CF = radii of the circle]
= 2√2 - 2
In square GIFH, GF is its diagonal. So we know that in a square,
diagonal = √2(Side)
GF = √2(GH)
GH = GF/√2
GH = (2√2 - 2) / √2 ∴ GH = 2 - √2
Hence, the side of the square is 2 - √2 cm
