question_answer

If two parallel chords on the same side of the centre of a circle are 6 cm and 8 cm and they are 1 cm apart, then the diameter of the circle (in cm) is

A)  14                         

B)         10  

C)  8                            

D)         5

Correct Answer: B

Solution :

 Let AB and CD be parallel chords of a circle (on the same side of the centre) of lengths \[6\,cm\] and \[8\,cm\] respectively. Let OLM be the perpendicular to the chords, with \[LM=1\text{ }cm\]. If r is the radius of the circle, then                 \[OL=\sqrt{{{r}^{2}}-{{4}^{2}}}\]                 \[OM=\sqrt{{{r}^{2}}-{{3}^{2}}}\] \[\therefore \]                  \[\sqrt{{{r}^{2}}-{{3}^{2}}}-\sqrt{{{r}^{2}}-{{4}^{2}}}=1\]                ??(i) Also        \[({{r}^{2}}-{{3}^{2}})-({{r}^{2}}-{{4}^{2}})=7\]                   .....(ii) Adding corresponding sides, we get                                 \[2\sqrt{{{r}^{2}}-{{3}^{2}}}=8\] or            \[{{r}^{2}}=25\] or            \[r=5\] Therefore diameter is \[10\text{ }cm.\]