ML Aggarwal Solution Class 9 Chapter 15 Circle Exercise 15.2

 Exercise 15.2

Question 1

If arcs APB and CQD of a circle are congruent, then find the ratio of AB: CD.

Sol :

⇒$\widehat{\mathrm{APB}}=\widehat{\mathrm{CQD}}$ (given)

∴AB=CD

(∵ If two arcs are congruent, then their corresponding chords are equal)

∴Ratio of AB and CD$=\frac{\mathrm{AB}}{\mathrm{CD}}=\frac{\mathrm{AB}}{\mathrm{AB}}=\frac{1}{1}$

⇒AB : CD = 1 : 1

Question 2

A and B are points on a circle with centre O. C is a point on the circle such that OC bisects ∠AOB, prove that OC bisects the arc AB.

Sol :

Given : In a given circle with centre O, A and B are two points on the circle. C is another point on the circle such that 

⇒∠AOC=∠BOC

To prove : arc AC= arc BC

Proof : ∵OC is the bisector of ∠AOB

or ∠AOC=∠BOC

⇒But these are the angle subtended by the arc AC and BC

∴arc AC= arc BC

(Q.E.D)

Question 3

Prove that the angle subtended at the centre of a circle is bisected by the radius passing through the mid-point of the arc.

Sol :

Given : AB is the arc of the circle with centre O and  C is the mid point of arc AB

To prove : OC bisects the ∠AOB

i.e. ∠AOC=∠BOC

Proof : ∵C is the mid point of arc AB

∴arc AC= arc BC

⇒But arc AC and arc BC subtend ∠AOC and ∠BOC at the centre

∴∠AOC=∠BOC

Hence OC bisects the ∠AOB 

(Q.E.D)

Question 4

In the given figure, two chords AB and CD of a circle intersect at P. If AB = CD, prove that arc AD = arc CB.

Sol :

Given : Two chords AB and CD of a circle intersects at P and AB=CD

To prove : arc AD=arc CB

Proof : AB=CD (given)

∴minor arc AB=minor arc CD

Subtracting arc BD from both sides

⇒arc AB-arc BD=arc CD-arc BD

⇒arc AD=arc CD

(Q.E.D)