# HOW TO FIND RADIUS WHEN LENGTH OF TWO PARALLEL CHORDS ARE GIVEN

## About "How to find radius when length of two parallel chords are given"

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,OB2  =  OE2 + EB2OB2  =  x2 + 52  ---(1) In triangle OFD,OD2  =  OF2 + FD2OD2 = (17-x)2 + 122 ---(2)

OB = OD (radius of the given circle)

(1)  =  (2)

x2 + 52  =  (17-x)2 + 122

x2 + 52  =  172 + x2 - 2(17) x + 122

x2 + 25  =  289 + x2 - 34x + 144

x2 - x2 + 34x + 25 - 144 - 289  =  0

34x - 408  =  0

34(x - 12)  =  0

x  =  12 cm

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

OB2  =  122 + 52

OB2  =  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,

 OB2  =  OP2 + PB2OB2  =  OP2 + PB252  =  OP2 + 32OP2  =  25 - 9OP2  =  16OP  =  √16OP  =  4 cm OD2  =  OQ2 + QD252  =  OQ2 + 4225  =  OQ2 + 16OQ2  =  25 - 16OQ2  =  9OQ  =  √9OQ  =  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,OA2  =  AP2 + PO252  =  42 + PO2 PO  =  √(25 - 16) PO  =  √9PO  =  3 cm In triangle COQ,OC2  =  OQ2 + CQ252  =  OQ2 + 32OQ  =  √(25 - 9)OQ  =  √16OQ  =  4 cm

PQ  =  PO + OQ

=  3 + 4

=  7 cm

Hence the length of PQ is 7 cm. After having gone through the stuff given above, we hope that the students would have understood "How to find radius when length of two parallel chords are given"

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