In the given figure, ∆ ABC is an equilateral triangle and radius of each smaller circle is 5 cm. Find the perimeter of ∆ABC.
Answer & explanation
Correct answer: option 2
In ∆ABC,
\(\angle\) ABC = 60° ( because ∆ABC is equilateral)
In ∆OBP,
\(\angle\) OBP = \(\frac{1}{2}\) x \(\angle\) ABC = 30°
Now,
In ∆OBP,
tan θ = \(\frac{Perpendicular}{Base}\)
⇒ tan 30° = \(\frac{1}{\sqrt {3}}\) = \(\frac{5}{BP}\)
BP = 5\(\sqrt {3}\)
CQ = BP = 5\(\sqrt {3}\)
BC = BP + PQ + QC = 5\(\sqrt {3}\) + 20 + 5\(\sqrt {3}\) = 10\(\sqrt {3}\) + 20
∆ABC is an equilateral triangle, therefore
AB =BC = CA
Perimeter = 3 x side = 3 x BC = 3 x (10\(\sqrt {3}\) + 20) = 30(\(\sqrt {3}\) + 2)