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NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 β Part 1
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:
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 5 cm and 11 cm respectively of a circle are parallel to each and are on opposite sides of its centre. If the distance between PQ and RS is 6 cm, find the radius of the circle
Solution:
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
- β¦ equation (1)
In right
- β¦ equation (2)
From equation (1) and (2)
Now, putting in equation (1)
So, the radius of the circle is .