If a line intersects two concentric circles with centre O at A, B, C and D, prove that AB = CD

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 given figure)

Let us draw a perpendicular OM on line AD.

It can be observed 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 centre of the circle bisects the chord.

∴ BM = MC ... (1)

And, AM = MD ... (2)

On subtracting equation (2) from (1), we obtain

AM − BM = MD − MC

⇒ AB = CD

Concept: Equal Chords and Their Distances from the Centre

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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.)

Given: A line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D.To Prove: AB = CD.Construction: Draw OM ⊥ BC.

Proof: ∵ The perpendicular drawn from the centre of a circle to a chord bisects the chord.∴ AM = DM ...(1)and BM = CM ...(2)Subtracting (2) from (1), we getAM - BM = DM - CM

⇒ AB = CD.

Solution

Given :

Two concentric circles with O. A line intersect them at A, B, C, and D

To prove: AB=CD

construction: Draw OM ⊥ AD, In bigger circle AD is chord OM ⊥ AD.

Proof

∴AM=MD [⊥ from centre of circle of a circle bisects the chord] ——–(i)

The smaller circle :

BC is chord OM ⊥ BC

BM=MC [⊥ from centre of the circle of a circle bisects the chord] ————–(ii)

On subtracting (i) from (ii)

AM-BM=MD-MC

AB=CD

Hence Proved

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In a figure, If a line intersects two concentric circlescircles with the same centre with centre O at A , B , C and D. Prove that AB = CD

Solution

Given: A line intersects two concentric circles (circles with the same center)
with center O at A, B, C, and D

To prove : AB = CD

Construction : Draw OM $⊥$ BC.

Proof: The perpendicular drawn from the centre of a circle to a chord bisects the chord. AM = DM ---(1) BM = CM -----(2) Subtracting (2) from (1), we get AM - BM = DM - CM
$\Rightarrow$ AB = CD

Hence Proved

Suggest Corrections

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.

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Last updated at March 2, 2017 by

Ex 10.4, 4 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 figure). Given: Two concentric circles with centre O. & a line intersects the circles at A,B,C & D To prove: AB = CD Proof: Let two circles be C1 & C2 and line be l We draw OP perpendicular to line l In circle C1, OP ⊥ BC So, OP bisects BC, i.e. BP = CP (As OP is perpendicular to line l ) (Perpendicular drawn from centre of a circle to a chord bisects the chord) In circle C2, OP ⊥ AD So, OP bisects AD, i.e. AP = DP (As OP is perpendicular to line l ) (Perpendicular drawn from centre of a circle to a chord bisects the chord) Subtracting (2) & (1), (2) – (1) AP – BP = DP – CP ⇒ AB = CD Hence proved

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Last updated at July 16, 2019 by Teachoo

Next: Ex 10.4, 6 Important →

Ex 10.4, 5 Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5 m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6 m each, what is the distance between Reshma and Mandip? Given: Let Reshma , Salma and Mandip be donated by points R,S & M resp. Given radius = 5 m & RS = 6m, SM =6m To find: Length of RM Construction: Join OR, OS, OM Let RM intersect OS at X Solution: In ORS & OMS OR = OM OS = OS RS = SM ORS OMS ROS = MOS In ORX & OMX OR = OM ROX = MOX OX = OX ORX OMX RX = MX Since RX = MX So, OX bisects chord RM, OX RM Now, Let OX = x, So, XS = OS OX = 5 x From (3) & (4) 25 x2 = 11 x2 + 10x 25 11 x2 + x2 = 10x 14 = 10x 10x = 14 x = 14/10 x = 1.4 Putting value of x in (3) RX2 = 25 x2 RX2 = 25 (1.4)2 RX2 = 25 1.96 RX2 = 23.04 RX = ("23.04" ) RX = 4.8 Therefore, RM = 2RX = 2 4.8 = 9.6 m Distance between Reshma and Mandip is 9.6 m