Question

if a line intersects two concentric circles with common centre O, at A,B,C and D.Prove that AB = CD​

Answer 1

$\underline{\underline{\bf{Given}}}$

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

$\underline{\underline{\sf{\bf{Required \: \: solution:}}}}$

As we know, OA= OD & OB = OC in the given respective circles.

In ∆OBC, we know OB = OC, so ∠OBC = ∠OCB

Therefore, ∠OCD = ∠OBA

In ∆OAD, we know that OA = OD, so ∠OAD = ∠ODA.

Since, ∠OCD = ∠OBA and ∠OAD = ∠ODA, we got ∠AOB in ∆OAB = ∠COD in ∆OCD.

So, from SAS (side angle side) congruency rule, we can say that ∆AOB ≅ ∆OCD.

Hence, AB = CD

___________________

Thanks!

Answer 2

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

As we know, OA= OD & OB = OC in the given respective circles.

In ∆OBC, we know OB = OC, so ∠OBC = ∠OCB

Therefore, ∠OCD = ∠OBA

In ∆OAD, we know that OA = OD, so ∠OAD = ∠ODA.

Since, ∠OCD = ∠OBA and ∠OAD = ∠ODA, we got ∠AOB in ∆OAB = ∠COD in ∆OCD.

So, from SAS (side angle side) congruency rule, we can say that ∆AOB ≅ ∆OCD.

Hence, AB = CD

___________________

Thanks!