Question

A carpenter wish to cut 4 equal circle from a circular piece of wood whose area is equal to 9pie cm^2. He want this circles of wood to be largest that can be possible, find the radius of each circles?

Answer 1

Let R be the radius of circular piece of wood

and r be the radius of smaller circles.

Now,OP=R , radius of larger circle

PB=r radius of smaller circle

OB=R−r

OA=OC    (Tangents drawn from same external points are equal)

∠OCB=∠OAB=90°     (Radius makes 90° angle with tangent)

⇒OA ∥ CB and OC ∥ AB

⇒OABC is a parallelogram

.⇒OA=BC and OC=ABOA=BC=r

Now,In △OAB , applying pythagoras theorem

OB2=OA2+AB2

(R−r)2=r2+r2R2−2rR+r2=2r2r2+2rR−R2=0r=−2R±4R2+4R2√2

=−2R±22√R2

=−R±2–√Rr=−R+2–√R   or  r=−R−2–√R

Radius can't be negative

So, r=−R+2–√R

Now,Area of larger circle=9π m2

πR2=9π

R=3 m

So,r=−3+32–√=3 (2–√−1) m.

Radius of each of the four new circle is 3 (2–√1)m

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