Question

f the length of the arc AB of this rocking toy chair is
14
π
cm and the angle of the sector formed by arc AB is
60
∘
, then what is the shortest length (in cm) between A and B?

Answer 1

Given :- the length of the arc AB of this rocking toy chair is

14π cm and the angle of the sector formed by arc AB is

60∘. Then what is the shortest length (in cm) between A and B ?

Solution :-

we know that,

• length of arc , which makes θ angle at centre and with radius as r cm is given by :- (θ/360) * 2 * π * r .

given that,

• Angle at centre = θ = 60° .
• Length of arc = 14π .
• radius = let r cm.

Putting values we get ,

→ (60/360) * 2 * π * r = 14π

→ (1/3) * πr = 14π

→ r = 14 * 3

→ r = 42 cm.

Now, in ∆AOB , we have ,

→ ∠AOB = 60° (given)

→ OA = OB = 42 cm. { Radius of circle.}

Therefore,

→ ∠OAB = ∠OBA = 60° . { Angle opposite to equal sides are equal.}

Therefore,

→ ∆OAB = An Equilateral ∆. { Each angle 60° .}

Hence,

→ OA = OB = AB { All sides of an equilateral ∆ are equal.}

→ AB = 42 cm. (Ans.)

∴ the shortest length (in cm) between A and B is 42 cm.

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