Two circles X and Y with centres A and B intersect at C and D. If area of circle X is 4 times area of circle Y, then AB=?

Question

This question is solved by taking angleACB = 90 in my book. How can we say that this angle a right angle triangle? Given answer is √5r.

Answer 1

You need information because you can put the circles so that it intersect and $AB$ can vary. I suppose that $\angle ACB =90^\circ$ is hypothesis.

Answer 2

By the triangle inequality $$AC+CB>AB$$ or $$2r+r>AB$$ or $$AB<3r.$$ Now, by the triangle inequality again we obtain $$AB+r>2r,$$ which gives $$r<AB<3r.$$