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Question

Find the perimeter of the minor segment ACB shown below.

A
2π+3 cm
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B
2π+6 cm
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C
π+12 cm
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D
π6 cm
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Solution

The correct option is B 2π+6 cm
Given, the radius of a circle (r)=OB=6 cm
and the angle subtentded by segment ACB at the centre of the circle(θ)=90.

Step 1: Let's find the length of AB:
In ODB, where ODAB and D is the mid point of AB
BOD=602=30

Using sin formula:
sin30=OppositeHypotenuse
12=BDOB (sin30=12)
Cross multiplying, we get
BD=OB×12BD=6×12BD=3 cm

AB=2×BD =2×3 =6 cm

Step 2: Length of the arc
BCA=θ360×π×d =60360×π×2×6
=16×π×12
=2π

Step 3: Perimeter of the segment ACB
=BCA+AB
=2π+6

Hence, the perimeter of the minor segment ACB is 2π+6 cm.
Option (b.) is the correct choice.

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