# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 1

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Q-1 Two circles of radii and intersect at two points and the distance between their centres is .Find the length of the common chord.

Solution:

Given, two circles with radius and .

• , and .

• Also (as we proved above)

• Let be .

In ,

• ………..equation (1)

In ,

• ………….equation (2)

Equating (1) and (2),

Putting the value of in (1) we get,

Therefore, length of the chord

Q-2 If two equal chords of a circle intersect within the circle; prove that the segments of one chord are equal to corresponding segments of the other chord.

Solution:

• Given, PQ and SR are chords intersecting at T and

• To prove, And

Construction, draw perpendicular bisectors of PQ and SR. Line from the center which bisects a chord is perpendicular to the chord.

• bisects

• bisects

As

• ……equation (1)

• Because M and N are midpoints of PQ and SR, ……equation (2)

In and

• (perpendiculars)

• (common line)

• ( and thus equidistant from the centre)

• By Right Angle Hypotenuse congruence condition.

• by Corresponding Parts of Congruent Triangles…….equation (3)

From (1) and (2) we get,

• (since we are adding equal parts (MT and TN) to equal quantities what we get according to Euclid is also equal)

• Therefore,

Again,

• (since we are subtracting equal parts (MT and TN) from equal quantities what is left according to Euclid is also equal)

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