how to find the perimeter of the shaded region in terms of π (pi) ? The diagram shows a quadrant with centre O and of radius, 9 cm. Given that the arc AM is 6 cm and the angleAOM is 2/3 rad , how...

how to find the perimeter of the shaded region in terms of π (pi) ?

The diagram shows a quadrant with centre O and of radius, 9 cm. Given that the arc AM is 6 cm and the angleAOM is 2/3 rad , how to find the perimeter of the shaded region in terms of π (pi) ?

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to write the formula of perimeter of shaded region to check what the problem provides and what you need to find, such that:

`P = MO+OB+MB`

Notice that `MO = OB = R = 9`  and you need to find the length of the arc MB.

You should remember that you may find the length of arc using the following formula such that:

`MB = pi*R*(hat(MOB))/180`

`hat(MOB) = pi/2 - 2pi/3 =gt hat(MOB) = (3pi-4pi)/6=gt hat(MOB) = -pi/6 =gt MB = -9pi/6`

Substituting -`9pi/6`  for MB in equation of perimeter yields:

`P = 9 + 9- 9pi/6 `

`P = 9(2- pi/6) =gt P = (9/6)(12 - pi) =gt P = (3/2)(12 - pi)`

Hence,evaluating the perimeter in terms of `pi` , using the given information, yields `P = (3/2)(12 - pi).`

Sources:
najm1947's profile pic

najm1947 | Elementary School Teacher | (Level 1) Valedictorian

Posted on

Required perimeter P= MO+OB+BM

MO=OB=Radius=9cm

angle AOB = pi/2 radian being angle of the quadrant

angle AOM = 2/3 radian

therefore angle MOB = pi/2-2/3

length of arc MB = angle*radius = (pi/2-2/3)*9 = 9pi/2-6 cm

P = 9+9+9pi/2-6 = 12+9pi/2 = 26.1cm appoximately

The perimeter of the shaded portion is 12+9pi/2 cm or 26.1cm approximately

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