Question

CHALLENGE FOR ALL MATH LOVING BRAINLIANS !!



GIVEN :
Three circles are placed such that they touch eachother and the diameter of each of the circle is 1 cm.
A band is surrounded such that it encloses the 3 circles.


TO FIND :
The length of the band.


FIGURE :
Refer the attachment.






No Spams and Copied Answers please !
​

Answer 1

$\large\underline{\sf{Solution-}}$

GIVEN :

Three circles are placed such that they touch eachother and the diameter of each of the circle is 1 cm.

A band is surrounded such that it encloses the 3 circles.

TO FIND :

The length of the band.

FIGURE :

Refer the attachment.

$\red{\large\underline{\sf{Calculations-}}}$

Since it is given that three circles are placed such that they touch eachother and the diameter of each of the circle is 1 cm.

Let assume that the center of three circles be A, B and C respectively and radius is r cm

So, triangle ABC is equilateral triangle.

So, ∠BAC = ∠ABC = ∠BCA = 60°.

Now, ID is a tangent to the circle with centre A and B.

And AD & BI are radii of the two circles,

We know, Radius and tangent are perpendicular to each other.

So, ∠ADI = 90° and ∠BID = 90°.

It means, AD || BI and also AD = BI = radii.

So, it implies, ABID is a parallelogram.

So, DI = AB = 2r = 1 cm. [ Opposite Sides of parallelogram]

Similarly, EF = HG = 1 cm

Now,

From figure, we concluded that ∠DAE = 120°.

So, length of arc DE is given by

$\boxed{ \rm{ Length \: of \: arc \: = \: 2\pi \: r \: \times \dfrac{ \theta}{360\degree } }}$

or

$\boxed{ \rm{ Length \: of \: arc \: = \: \pi \: d\: \times \dfrac{ \theta}{360\degree } }}$

where, d is diameter of circle and theta is central angle.

So, on substituting the values, we get

$\rm :\longmapsto\: Length \: of \: arc , \: DE\: = \: \pi \: (1)\: \times \dfrac{ 120\degree }{360\degree }$

$\bf :\longmapsto\: Length \: of \: arc , \: DE\: = \dfrac{ \pi }{3}$

Similarly,

$\bf :\longmapsto\: Length \: of \: arc , \: FG\: = \dfrac{ \pi }{3}$

and

$\bf :\longmapsto\: Length \: of \: arc , \: HI\: = \dfrac{ \pi }{3}$

So, total length of the band =

Length of arc DE + EF + Length of arc FG + GH + Length of arc HI + ID

$\rm \: = \: \: \dfrac{\pi}{3} + 1 + \dfrac{\pi}{3} + 1 + \dfrac{\pi}{3} + 1$

$\rm \: = \: \: 3 + \pi$

$\rm \: = \: \: 3 + \dfrac{22}{7}$

$\rm \: = \: \: \dfrac{21 + 22}{7}$

$\rm \: = \: \: \dfrac{43}{7} \: cm$

Remark :-

Short Cut Trick :-

• Length of band = Circumference + 3 diameter

Answer 2

Answer:

Happy Birthday yrr be happy always gbu

Sorry