NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 2

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Circle with center O and chords AB and CD are of equal length

Circle With Center O and Equal Chords

Circle with center O and chords AB and CD are of equal length

Q-3 If two equal chords of a circle intersect within the circle; prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Solution:

in Circle With Center O With AB and PQ Are Two Chords

Circle and its chord AB and PQ intersecting at S,CD is the diameter.

Given,

  • AB and PQ are chords intersecting at S.

  • , CD is the diameter.

To prove,

Now, Construction,

  • Draw And OS is joined.

In and ,

  • (Equal chords are equidistant from the centre)

  • (Common line)

  • ( by construction)

  • Therefore, (By Right Angle Hypotenuse congruence condition)

Therefore,

  • by Corresponding Parts of Congruent Triangles

  • (Same angle)

Q-4 If a line intersects two concentric circles (circles with the same centre) with centre O. A common chord intersects the circles at A, B, C and D, prove that AB = CD (see Fig.).

Two Concentric Circles With Centre O at a,B,C and D

Two concentric circles with centre O and common chord intersecting at A,B,C and D

Solution:

Given,

Two concentric circles, with centre at O and common chord intersecting at A,B,C and D

Contruction, draw a line OM ⊥ AD, intersecting AD at M.

Two Concentric Circle With Centre at O and Common Chord Intersecting at a,B,C and D

Two concentric circle with centre O and common chord intersecting at A,B,C and D also OM ⊥ AD. OM bisects AD as OM ⊥ AD

Now, since and is drawn from center O

  • OM is bisects of AD as .

  • …………equation (1)

Since O is also the center of inner circle, therefore OM also bisects BC as .

  • Therefore, …….equation (2)

From (1) and (2),

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