NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 1

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Theorems of the 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 RS 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 RS 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 .