Maths / Circles / Perpendicular From Centre of Circle Bisects the Chord and Vice Versa

QUESTION

 In the given figure, O is the centre of the circle of radius 5 cm. If OM = 4 cm, find the length of PN.
EXPLANATION
Explain TypeExplanation Content
Text

OM = 4 cm

$\fn_jvn \large Now,\: \Delta OMN\: is\: a\: triangle,\: where\: \angle =90^0,\: r =ON=5cm\: and\: OM=4 cm.$

By pythagoras theorem,

$\fn_jvn \large ON^2=OM^2+MN^2\\\\ \Rightarrow \: \: MN^2=ON^2-OM^2\\\\ \: \: \: \: \: r^2-4^2\\\\ 5^2-4^2\\\\ \Rightarrow \: \: MN =\sqrt{25-16}=\sqrt{9}=3\: cm\\\\ Also,\: OM\perp PN\\\\ \Rightarrow \: \: PM=MN=\frac{1}{2}PN [Perpendicular \:\: \, draw\: \, from\:\, the \, centre\: of\:\, a\: \, chord\:\, bisects\:\, the\: \, chord]\\\\ \Rightarrow \: \: PN=2MN\Rightarrow PN=2(3)=6cm$

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