**How to find radius when length of two parallel chords are given ?**

Here we are going to see some examples problems on finding radius when length of two parallel chords are given.

**Example 1 :**

AB and CD are two parallel chords of a circle which are on either sides of the centre. Such that AB = 10 cm and CD = 24 cm. Find the radius if the distance between AB and CD is 17 cm.

**Solution :**

Consider the right triangles OEB and OFD,

In triangle OEB, OB OB |
In triangle OFD, OD OD |

OB = OD (radius of the given circle)

(1) = (2)

x^{2} + 5^{2} = (17-x)^{2} + 12^{2}

x^{2} + 5^{2} = 17^{2 }+ x^{2} - 2(17) x + 12^{2}

x^{2} + 25 = 289^{ }+ x^{2} - 34x + 144

x^{2} - x^{2} + 34x + 25 - 144 - 289 = 0

34x - 408 = 0

34(x - 12) = 0

x = 12 cm

By applying the value of x in the 1^{st} equation, we get

OB^{2} = 12^{2} + 5^{2}

OB^{2} = 144 + 25 = 169

OB = √169 = 13 cm

**Example 2 :**

In the figure given below, AB and CD are two parallel chords of a circle with centre O and radius 5 cm such that AB = 6 cm and CD = 8 cm. If OP ⊥ AB and CD = OQ determine the length of PQ.

**Solution :**

Here we have two right triangles,

Triangle OPB and triangle OQD.

OB = OD = radius of the circle = 5 cm

In Δ OPB,

OB OB 5 OP OP OP = √16 OP = 4 cm |
OD 5 25 = OQ OQ OQ OQ = √9 OQ = 3 cm |

PQ = OP - OQ

= 4 - 3

= 1 cm

Hence the length of PQ is 1 cm

**Example 3 :**

In the figure given below, AB and CD are two parallel chords of a circle with centre O and radius 5 cm. Such that AB = 8 cm and CD = 6 cm. If OP = AB and OQ ⊥ CD.determine the length PQ.

**Solution : **

Consider the triangles APO and COQ

OA = OC = radius of the circle = 5 cm

AP = PB = 4 cm

CQ = QD = 3 cm

In triangle APO, OA 5 PO = √(25 - 16) PO = √9 PO = 3 cm |
In triangle COQ, OC 5 OQ = √(25 - 9) OQ = √16 OQ = 4 cm |

PQ = PO + OQ

= 3 + 4

= 7 cm

Hence the length of PQ is 7 cm.

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