NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 2

Download PDF of This Page (Size: 240K)

Angle subtended by the segment at center is twice the angle subtended by the same segment at cicumference

Angle Subtended by the Segment at Center and at Cicumference

Angle subtended by the segment at center is twice the angle subtended by the same segment at cicumference

Q-3 The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?

Solution:

Points P, Q, R, and S Are Concyclic

Points P, Q, R, S, on circle, also chords PQ and RS, AO⊥PQ and OB⊥RS, PQ∥RS

Given,

  • PQ and RS are two parallel chords of a circle with centre O

  • , and

  • The radius of the circle is .

Since and and , therefore O, Q, and P are collinear.

Since perpendiculars from center bisect the chord, therefore

So, the distance of chord RS from the centre is ( )

Q-4 Let the vertex of an angle QPR be located outside a circle and let the sides of the angle intersect equal chords QA and SR with the circle. Prove that is equal to half the difference of the angles subtended by the chords QR and AS at the centre.

Solution:

PQR Is a Triangle With Vertex P Located Outside a Circle

PQR is a triangle with vertex P located outside a circle, the sides of the angle intersect equal chords OA and RS.

In a triangle sum of external angle is equal to the sum of interior opposite angles. Therefore, in , we have

…………equation (1)

Also angle subtended at the centre is twice the angle at any point on the remaining part of circle, therefore,

  • ………equation (2)

From (1) and (2), we have

  • ( )

So, is equal to half the difference of angles subtended by the chords QR and AS at the centre.