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In an equilateral triangle, three identi...

In an equilateral triangle, three identical coins of radii 1 units each, are kept so that they touch each other and also the sides of a triangle. Find the area of the triangle.

Text Solution

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The correct Answer is:
`(4sqrt3+6)` sq. unit
The given arrangement of coins is as shown in the given figure.

To find the area of the triangle, we need to find its side.
For circle with centre `C_2`, BP and BP' are two tangents from B to circle,
Therefore, `BC_(2)` must be angle bisector of `angleB`.
`angle B=60^@" "( :.DeltaABC" is equilateral")`
`:. angleC_(2)BP=30^(@)`
`:. tan30^(@)=1/x`
`rArr x=sqrt3`
`:. BC=BP+PQ+QC`
`=x+2+x=2+2sqrt3`
Area of an equilateral triangle is `sqrt3/4 a^2`, where a is side length.
`:. "Area of " DeltaABC=sqrt3/4(2+2sqrt3)^2`
`=sqrt3(1+3+2sqrt3)`
`=(4sqrt3+6)` sq. unit
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