Two parallel chords of a circle, which are on the same side of the centre, subtend angles 72 and 144 degree respectively at the centre. Prove that the perpendicular distance between the chords is half the radius of the circle

Asked by Anil | 20th Jun, 2017, 07:06: PM

Expert Answer:

NOTE: The image is drawn without accurate measurement. 
begin mathsize 16px style In space the space diagram space we space can space see space angle COD equals 72 degree space and space OP space is space perpendicular space to space AB comma space OQ space is space perpendicular space to space CD
Also space in space increment OCD comma
OC equals OD space space space space space space space space space space space space space Radius space of space the space same space circle
rightwards double arrow angle OCD equals angle ODC space space space space Angles space of space opposite space sides space which space are space equal
rightwards double arrow angle OCD equals angle ODC equals straight x
rightwards double arrow angle OCD plus angle ODC plus angle COD equals 180 degree
rightwards double arrow straight x plus straight x plus 72 degree equals 180 degree
rightwards double arrow straight x equals angle OCD equals angle ODC equals 54 degree
Similarly space you space can space take space increment OAB space and space prove space that space angle OAB equals angle OBA equals 18 degree
sin 18 degree equals OP over straight r space
OP equals straight r cross times sin 18 degree space.... left parenthesis straight i right parenthesis
sin 54 degree equals OQ over straight r
OQ equals straight r cross times sin 54 degree...... left parenthesis ii right parenthesis
Subtracting space left parenthesis ii right parenthesis space from space left parenthesis straight i right parenthesis
OP minus OQ equals straight r open parentheses sin 18 degree space minus sin 54 degree close parentheses
PQ equals straight r over 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses sin 18 degree space minus sin 54 degree equals 0.5 equals 1 half close parentheses


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Answered by Sneha shidid | 21st Jun, 2017, 10:16: AM