# Ab and Cd Are Two Parallel Chords of a Circle Such that Ab = 24 Cm and Cd = 10 Cm. If the Radius of the Circle is 13 Cm. Find the Distance Between the Two Chords. - Mathematics

AB and CD are two parallel chords of a circle such that AB = 24 cm and CD = 10 cm. If the
radius of the circle is 13 cm. find the distance between the two chords.

#### Solution

Join OA and OC.

Since the perpendicular from the centre of the circle to a chord bisects the chord.

Therefore, N and M are the mid-points of AB and CD respectively.

Consequently

AN = NB = 1/2 AB = 1/2 xx 24 = 12 cm and

CM = MD = 1/2 CD = 1/2 xx 10 = 5 cm

In right-angled triangles ANO and CMO, we have

OA2  = ON2 + AN2 and OC2 = OM2 + CM2

⇒ 132 = ON2 + 122  and 132 = OM2 + 52

⇒ ON2 = 132 - 122 and OM2 = 132 - 52

⇒ ON2 = 169 - 144  and OM2 = 169 - 25

⇒ ON2 = 25  and OM2 = 144

⇒ ON = 5 and OM = 12

Now, NM = ON + OM = 5 + 12 = 17cm

Hence, the distance between the two chords is 17 cm.

Concept: Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)
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