In the figure given below, O is the centre of the circle. AB and CD are two chords of the circle. OM is perpendicular to AB and ON is perpendicular to CD.

AB = 24 cm, OM= 5 cm, ON= 12 cm. Find the:

1) radius of the circle

2) length of chord CD.

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#### Solution

A line from centre to a chord that is perpendicular to it, bisects it.

It is given that AB = 24 cm

Thus, MB = 12 cm

1) Applying Pythagoras theorem for ΔOMB,

`OM^2 + MB^2 = OB^2`

`=> 5^2 + 12^2 = OB^2`

`=> OB = 13`

Thus, radius of the circle = 13 cm.

2) Similarly, applying Pythagoras theorem for ΔOND,

`ON^2 + ND^2 = OD^2`

OD is the radius of the circle

`=> 12^2 + ND^2 = 13^2`

`=> ND = 5`

A line from center to a chord that is perpendicular to it, bisects it.

ND = 5 cm

Thus, CD = 10 cm

Concept: Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal

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