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)