Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal

Solution:

Using equal angles at the centers and the fact that circles are congruent, we prove the statement using Side-Angle-Side (SAS criteria) and corresponding parts of congruent triangles (CPCT).

Draw chords QR and YZ in two congruent circles as shown above. Join the radii PR, PQ, and XY, XZ respectively.

Given that chords subtend equal angles at the center. So, ∠QPR = ∠YXZ.

We need to prove that chords are equal, that is, QR = YZ

Since the circles are congruent, their radii will be equal.

PR = PQ = XZ = XY

Consider the two triangles ∆PQR and ∆XYZ.

PQ = XY (Radii are equal)

∠QPR = ∠YXZ (Chords subtend equal angles at center)

PR = XZ (Radii are equal)

By SAS criteria, ∆PQR is congruent to ∆XYZ.

So, QR = YZ (Corresponding parts of congruent triangles)

Hence proved if chords of congruent circles subtend equal angles at their center then the chords are equal.

☛ Check: Class 9 Maths NCERT Solutions Chapter 10

Video Solution:

Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Maths NCERT Solutions Class 9 Chapter 10 Exercise 10.2 Question 2

Summary:

If chords of congruent circles subtend equal angles at their centers, then the chords are equal.

☛ Related Questions:

• Fill in the blanks:(i) The center of the circle lies in____of the circle. (exterior / interior)(ii) A point, whose distance from the center of circle is greater than its radius lies in____of the circle. (exterior / interior)(iii) The longest chord of the circle is a_____of the circle.iv) An arc is a______when its ends are the ends of a diameter.(v) Segment of a circle is the region between an arc andof the circle.(vi) A circle divides the plane, on which it lies, inparts.
• Write True or False: Give reasons for your answers.(i) Line segment joining the centre to any point on the circle is a radius of the circle.(ii) A circle has only finite number of equal chords.(iii) If a circle is divided into three equal arcs, each is a major arc.(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.(v) Sector is the region between the chord and its corresponding arc.(vi) A circle is a plane figure.
• Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centers.