NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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

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

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)