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

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PQR and ABC is triangle ,∠B=∠Q=90,AB=PQ or BC=QR

RHS Congruence Rule

PQR and ABC is triangle ,∠B=∠Q=90,AB=PQ or BC=QR

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:

Two Circle With Radius 5cm and 3cm

Two circle with radii 5cm and 3cm respectively, OP = 5cm, PS =3 cm and OS=4cm

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

in Circle PQ and SR Are Two Chords

In circle PQ and SR are two Chords, intersecting at T

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)