I was going thru this question – Two Chords Ab, Cd of Lengths 5 Cm, 11 Cm Respectively of a Circle Are Parallel. If the Distance Between Ab and Cd is 3 Cm, Find the Radius of the Circle.
As per explanation provided in here I disagree – how can one be sure that cords are on one side of the semi-circle. I was trying the same method but judged that the chords are in each other hemisphere.
That means that the radius can vary based on distance between the chords if they are (i) on the same side of the center. (ii) on the opposite sides of the center.
In fact we get wrong answer if they are in opposite center – so how can we know from an approach when to consider chords on same / other side.
Best Answer
I would say it is not immediately clear that the two chords should be on the same side of the center. So you are right to prove that indeed it is impossible that they are on opposite sides of the center.
One way to prove this is to note that the radius $r$ of the circle satisfies $r\geq\tfrac{11}{2}$, because the circle has a chord of length $11$. Then by Pythagoras the distance from the chord of length $5$ to the center is $$\sqrt{r^2-\left(\tfrac52\right)^2}\geq\sqrt{\frac{11^2-5^2}{2^2}}=2\sqrt{6}>3,$$ so if the distance to the other chord is $3$ then it cannot be on the other side of the center.