NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Theorems of the circle

Theorems of Circle

Q-1 Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Solution:

Two Circles with Centres P and R

Given

  • Two circles with centres P and R, which intersect each other at Q and S

Construction

  • Join PQ, PS, SR, QR.

In and

  • (radius of the same circle)
  • (radius of the same circle)
  • (common line)
  • Therefore by Side-Side-Side criterion of congruence,

Therefore, (corresponding parts of congruent triangles)

Q-2 Two chords PQ and ₹ of lengths 5cm and 11cm respectively of a circle are parallel to each and are on opposite sides of its centre. If the distance between PQ and ₹ is 6cm, find the radius of the circle

Solution:

O is the Centre of Circle

Given,

  • O is the centre of the circle
  • And its radius be r cm
  • and

Construction,

  • Join OP and OR
  • Since both P and R are on the circle and O is the center,

and and , Therefore, B, O, A are collinear.

Let , since . Therefore,

Since, the perpendicular from the centre to a chord of the circle bisects the chord, therefore,

  • ()
  • ()

In right

  • … equation (1)

In right

  • … equation (2)

From equation (1) and (2)

Now, putting in equation (1)

So, the radius of the circle is .