If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 10.25).

Solution:

Draw a perpendicular from the center of the circle OM to the line AD.

We can see that BC is the chord of the smaller circle, and AD is the chord of the bigger circle.

We know that perpendicular drawn from the center of the circle bisects the chord.

∴ BM = MC ... (1)

and, AM = MD ... (2)

Subtracting (2) from (1), we obtain

AM − BM = DM − CM

∴ AB = CD

☛ Check: NCERT Solutions for Class 9 Maths Chapter 10

Video Solution:

If a line intersects two concentric circles (circles with the same center) with center O at A, B, C and D, prove that AB = CD (see Fig. 10.25).

NCERT Solutions Class 9 Maths Chapter 10 Exercise 10.4 Question 4

Summary:

If a line intersects two concentric circles (circles with the same center) with center O at A, B, C, and D, then AB = CD. 

☛ Related Questions:

• If two circles intersect at two points, prove that their centers lie on the perpendicular bisector of the common chord.
• Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centers is 4 cm. Find the length of the common chord.
• 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.
• If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the center makes equal angles with the chords.