The correct answer is C

Name the points as shown

Draw \(OH\perp BC\),then H is the midpoint of BC.So BH = 1 cm

Quadrilateral ABHK has 3 right angles at A,B and H

=> ABHK is a rectangle => HK = AB = 2 cm 

Let r be the radius of the circle,so OH = KH - OK = 2 - r (cm)

\(\Delta OHB\) right at H has : OH² + HB² = OB² (Pythagoras theorem)

\(\Rightarrow\left(2-r\right)^2+1^2=r^2\Rightarrow4-4r+r^2+1-r^2=0\)
\(\Rightarrow5-4r=0\Rightarrow r=\dfrac{5}{4}\)

Name the points as shown

Draw OH⊥BC

,then H is the midpoint of BC.So BH = 1 cm

Quadrilateral ABHK has 3 right angles at A,B and H

=> ABHK is a rectangle => HK = AB = 2 cm 

Let r be the radius of the circle,so OH = KH - OK = 2 - r (cm)

ΔOHB

 right at H has : OH2 + HB2 = OB2 (Pythagoras theorem)

⇒(2−r)2+12=r2⇒4−4r+r2+1−r2=0

⇒5−4r=0⇒r=54

We have  \(AB=AM+BM=10+6=16;BC=BN+CN=6+x;CA=CP+AP=x+10\).

The triangle ABC has his edges   \(a=10+6=16;b=6+x;c=x+10\) . Call  \(p\)  is the halp perimeter of \(ABC\) then  \(p-a=10,p-b=6,p-c=x\) and the area of ABC is

                                      \(\sqrt{p\left(p-a\right)\left(p-b\right)\left(p-c\right)}=\sqrt{60x\left(16+x\right)}\) .

By the assumptions we have    \(60x\left(x+16\right)=\left(120\sqrt{3}\right)^2\Leftrightarrow x^2+16x-720=0\Leftrightarrow x\in\left\{20;-36\right\}\).

So, \(x=20;p=36\), the perimeter of  ABC is  72.

Suppose CH is the height of CAB, then \(CH=\dfrac{2.120\sqrt{3}}{16}=15.\)  The triangle CHA has

                                  \(CHA=90^0;CH=15;CA=30\Rightarrow A=60^0\). 

The measure of angle A is    \(60^0\) .

We have  AB=AM+BM=10+6=16;BC=BN+CN=6+x;CA=CP+AP=x+10

.

The triangle ABC has his edges   a=10+6=16;b=6+x;c=x+10

 . Call  p  is the halp perimeter of ABC then  p−a=10,p−b=6,p−c=x

 and the area of ABC is

                                      √p(p−a)(p−b)(p−c)=√60x(16+x)

 .

By the assumptions we have    60x(x+16)=(120√3)2⇔x2+16x−720=0⇔x∈{20;−36}

.

So, x=20;p=36

, the perimeter of  ABC is  72.

Suppose CH is the height of CAB, then CH=2.120√316=15.

  The triangle CHA has

                                  CHA=900;CH=15;CA=30⇒A=600

. 

The measure of angle A is    600

 .