Question

Area of triangle if radius of the circle is 1cm.

a)" "4(sqrt3)

b)" "7 + 4(sqrt3)

c)" "4(sqrt3) - 6

c)" "6 + 4(sqrt3)

Solution

In the below triangle we can see that there are three similar circles arranged.

Since radius of all circles are equal to 1. So, OP = PQ = QO = 2cm.
Hence triangle QOP are equilateral trianle.

Now angle QOP = 60 degree, so angle TOR = 120 degree.

In the quadilateral BTOR, OT and OR are perpendicular to side AB and BC.

Angle TBR + 90 + 90 + 120 = 360

=> Angle TBR = 60 degree.

Similarly we can prove for other angle as well.

Hence triangle ABC is equilateral triangle.

Now,

Radius of the circle = 1cm = OR

Also tan30 = (OP)/(BR)

=> BR = (OP)/tan30 = 1/(1/sqrt3) = sqrt3

Similarly, SC = sqrt3

Also we can see that OP = 1 + 1 = 2

and OP = RS

Hence side of triangle = BC = BR + RS + SC =sqrt3 + 2 + sqrt3 = 2 + 2sqrt3

We know that area of equlateral triangle = ((sqrt3)/4)xx(Side)^2 = ((sqrt3)/4)xx(2 + 2sqrt3)^2 = 6 + 4sqrt3