What is the length of arc ACB?

Question

PA & PB are the tangents of the circle with center 0 and circumference 12cm. What is the length of arc ACB?

A. 3
B. 4
C. 5
D. 8
E. 10

CrackverbalGMAT · Answer

Triangles POA and POB are congruent triangles by the RHS

\angle{PAO} = \angle{PBO} = 90^o  (Angle formed between the radius and the tangent)

PO is the hypotenuse, which is the same for both triangles

OA = OB = Radius.

Since they are congruent,  \angle{APO} = \angle{BPO} = 30^o  and therefore we get  \angle{POA} = \angle{POB} = 60^o

The central angle  \angle{AOB} = 120^o

Length of arc =  \frac{Central \space Angle}{360} * Circumference = \frac{120}{360} * 12 = 4

Option B

Arun Kumar

QuantMadeEasy · Answer

Step 1: Understanding the question 
C = 2 π r = 12

As PA & PB are the tangents, ∠A = ∠B= 90
Refer to the attachment
∠P + ∠A + ∠AOB + ∠B = 360
60 + 90 + ∠AOB + 90 = 360
∠AOB = 120

arc length ACB = (120/360) * 2 π r = (1/3) * 12 = 4

B is correct

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