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Perpendicular from the Center to a Chord Bisects the Chord

In the given ...

Question

In the given figure, O is the centre of the circle with radius 5 cm. OP and OQ are perpendiculars to AB and CD respectively. AB = 8 cm and CD = 6 cm. Determine the length of PQ.

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Solution

Given: $Radius = 5 \ cm, \ OP \perp AB \ and \ OQ \perp CD$

$AB = 8 \ cm \ and \ CD = 6 \ cm$

$P \ and \ Q$ are mid point of $AB \ and \ CD$ respectively ( $\perp$ from the center to a chord will bisect the chord)

In $\triangle OAP$

$OA^2= OP^2+AP^2$

$25 = OP^2 + 16 \Rightarrow OP = 3 \ cm$

Similarly, in $\triangle OCQ$

$OC^2= OQ^2+CQ^2$

$25 = OQ^2+9 \Rightarrow OQ = 4 \ cm$

Hence $PQ = OP + OQ = 3 + 4 = 7 \ cm$

$\\$

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