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Question
The perimeter of an arc of radius 4.2 cm is 12.8 cm. Determine the angle subtended by the arc at the centre of circle.
Solution
Given: Radius of arc = 4.2 cm and perimeter of arc = 12.8 cm.
Consider the sector OACB,
Length of the arc ABC = `θ/360^circ xx 2 πr` ......(i)
Perimeter of the sector OACB = OA + arc ACB + OB
= `r + θ/360 xx 2πr + r`
= `2r + θ/360 xx 2πr`
Thus, the perimeter of the arc = `2r + θ/360 xx 2πr`
According to the question,
`2r + θ/360 xx 2πr` = 12.8 ......(ii)
Substituting the value of r in equation (ii),
`2 xx 4.2 + θ/360 xx 2π xx 4.2` = 12.8
`8.4 + θ/360 xx 2π xx 4.2` = 12.8
`θ/360 xx 2π xx 4.2` = 12.8 – 8.4
`θ/360 xx 2 xx 22/7 xx 4.2` = 4.4
`θ/360 xx 2 xx 22 xx 0.6` = 4.4
θ = `(4.4 xx 360)/(22 xx 0.6 xx 2)` = 60°
As a result, the circle's center's arc subtends a 60° angle.
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