Question

The perimeter of a certain sector of a circle is equal to the length of the area of the semi-circle
having the same radius, express the angle of the sector in degrees, minutes and seconds.
​

Answer 1

Correct Question :-

The perimeter of a certain sector of a circle is equal to the length of the arc of the semi-circle having the same radius, express the angle of the sector in degrees, minutes and seconds.

Sᴏʟᴜᴛɪᴏɴ :-

Let us Assume That, radius of sector and radius of semi-circle is r cm and angle at centre of sector is @.

Than,

→ Perimeter of sector = Arc of semi - circle

→ {(@/360°) * 2 * π * r } + 2r = (1/2) * 2 * π * r

→ 2r{(@/360°)π + 1} = πr

→ (@/360°) * π + 1 = π/2

Putting π = 180° Now,

→ (@/360°) * 180° + 1 = 180°/2

→ @/2 + 1 = 90°

→ (@ + 2) = 180°

→ @ = 180° - 2

→ @ = (π - 2)

→ @ = (3.14 - 2)

→ @ = 1.14 radians.

Now,

→ π radian = 180°

→ 1.14 radian = (180/π) * 1.14 ≈ 65.3503°.

Now,

→ 1° = 60 min.

→ 0.3503° = 60 * 0.3503 = 21.018Min.

Now,

→ 1Min. = 60 seconds .

→ 0.18 Min = 60 * 0.18 ≈ 11 seconds.

Hence, The angle of the sector is 65°21'11".

Answer 2

${ \bold{ \huge{ \underline{ \underline{ \blue{Question:-}}}}}}$



▪ The perimeter of a certain sector of a circle is equal to the length of the area of the semi circle having the same radius. Express the angle of the sector in degrees, minutes and seconds.



${ \huge{ \bold{ \underline{ \underline{ \blue{Solution:-}}}}}}$



▪ Let.....



➧Radius of sector = radius of semicircle =r cm



➧angle at the centre of the sector = α



${ \bold{ \red{ GIVEN- }}}$



✿ Perimeter of sector = length of arc of semi
circle



${ \bold{ \red{TO \: FIND- }}}$



✿ Angle of the sector in degrees,minutes and seconds



${ \boxed{ \bold{ \pink{(( \frac{ \alpha }{360} ) \times 2 \: \pi \: r \: + 2r)}} = { \green{ \frac{1}{2} \times 2 \: \pi \: r}} }}$



${ \bold{ \implies{ \pink{2r(( \frac{ \alpha }{360} ) \: \pi \: + 1)}} = { \green{\pi \: r}}}}$



${ \bold{ \implies{ \pink{2(( \frac{ \alpha }{360} )\pi \: + 1)}} = { \green{\pi}}}}$



${ \bold{ \implies{ \pink{2(( \frac{ \alpha }{360} \times 180) + 1)}} = { \green{180}}}}$



${ \bold{ \implies{ \pink{( \frac{ \alpha }{2} + 1)}} = { \green{ \frac{180}{2}}}}}$



${ \bold{ \implies{ \pink{ \alpha + 2}} = { \green{ \frac{180}{2} \times 2}}}}$



${ \bold{ \implies{ \pink{ \alpha + 2}} = { \green{180}}}}$



${ \boxed{ \bold{ \implies{ \red{ \alpha = 180 - 2}}}}}$



${ \boxed{ \bold{ \implies{ \red{ \alpha = \pi \: - 2}}}}}$



${ \bold{ \implies{ \alpha \: = 3.14 - 2}}}$



${{ \boxed{ \bold{ \implies{ \red{ \alpha = 1.14 \: radians \: \: }}}}}}$



➺ π radian = 180°


➺1.14 radian = 180°/ π × 1.14 = 65. 3503° ✔




➺ 1° = 60 minutes


➺65.3503° = 0.3503×60 = 21.018 min ✔




➺ 1 min = 60 seconds


➺0.18 min = 0.18×60 = 11 seconds ✔



therefore,



${ \bold{ \underline{ \red{the \: angle \: of \: sector }}}}$


= 65° 21' 11"