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Question

$$O$$  is the centre of the circle with radius  $$5cm.$$  Chords  $$AB$$  and  $$C D$$  are parallel.  $$A B = 6 cm$$  and  $$C D = 8cm.$$  If  $$P Q$$  is distance between  $$A B$$  and  $$C D ,$$  then find  $$PQ.$$
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Solution

$$Applying\, \, Pythagoras\, \, theorem\, \, in \\ \, \angle AOP\, \, and\, \, \Delta COQ$$

$$ \\ In\, \, \angle AOP, \\ A{ O^{ 2 } }=A{ P^{ 2 } }+P{ O^{ 2 } } \\ \Rightarrow A{ O^{ 2 } }={ \left( { A{ B^{ 2 } } } \right) ^{ 2 } }+P{ O^{ 2 } } \\ { \left( 5 \right) ^{ 2 } }={ \left( 4 \right) ^{ 2 } }+{ x^{ 2 } } \\ \Rightarrow 25-16={ x^{ 2 } } \\ \therefore x=\sqrt { 9 } =3\, \, cm$$

$$ \\ In\, \, \, \Delta COQ, \\ C{ O^{ 2 } }=O{ Q^{ 2 } }+C{ Q^{ 2 } } \\ \Rightarrow C{ O^{ 2 } }=O{ Q^{ 2 } }+{ \left( { C{ D^{ 2 } } } \right) ^{ 2 } } \\ \Rightarrow 25-9={ y^{ 2 } } \\ \Rightarrow y=\sqrt { 16 } =4\, \, cm$$

$$ \\ Therefore\, \, ,PQ=PO+OQ=3+4=7\, \, cm$$

Mathematics

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