Question

In the figure find the length of an arc AB of a circle centre O if AOB=90°

Answer 1

Answer:(1) In the given figure, OA and OB are the radii of the circle. OA=OB=AB (Given) ∴△OAB is an equilateral triangle. ⇒∠AOB=∠OAB=∠OBA=60 o Thus, the measure of ∠AOB is 60 o . (2) The measure of an angle subtended by an arc at a point on the circle is half of the measure of the angle subtended by the arc at the centre. ∠ACB= 2 1​∠AOB= 2 1​60o=30 o Thus, the measure of ∠ACBis30 o . (3) m(arcAB)=∠AOB=60 o (Measure of an arc is the measure of its corresponding central angle) Thus, the measure of arc AB is 60 o . (4) m(arcACB)=360 o - m(arcAB)=360 o −60 o =300 o Thus, the measure of arc ACB is 300 o .

Answer 2

Given:

Arc AB which subtends angle 90° at center.

To Find:

Length of arc

Solution:

As the length of the arc is given by,

= Ф/360 × 2πr

= Ф/180 × πr   (where r is the radius of the circle.)

AOB = 90 ∴  Ф = 90°

put  Ф = 90° in the formula of the length of the arc

⇒  Ф/180 × πr

⇒ 90/180 × πr

⇒ πr/2

Hence, the length of the arc is  πr/2.