In the adjoining figure, `AC` is the diameter of a circle with centre `O` and chord ` BD _|_ AC` , intersecting each other at `E`. Find out the values of `p,q,r` in terms of `x`.

Explanation

(i) Arc `AD` subtends `/_AOD` at the centre and `/_ ABD` at a point `B` on the remaining part of the circle.
Now,` /_ AEB = 90^(@) = ( x^(@))/(2)`.
Now, `/_AEB = 90^(@) [ :' BD _|_ AC ]`.
From `Delta AEB`,we get
`p^(@) = 180^(@) - ( /_ ABE + /_ AEB )`
`implies p^(@) = 180^(@) - ( /_ ABD + /_ AEB ) = 180^(@) -((x)/(2)+90^(@)) = (90^(@) - (x^(@))/(2))`
`:. p = 90 - (x)/(2)`.
(ii) Arc AD subtends `/_ AOD` at the centre and `/_ ACD ` at a point C of the circle.
`:. /_ ACD = (1)/(2) /_ AOD = (x^(@))/( 2) implies q = (x)/(2)`
(iii) `/_ ABC = 90^(@) `[ angle in a semicircle ]
`implies /_ ABE + /_ CBE = 90^(@) implies /_ ABD +/_CBE = 90^(@)`
`implies (x^(@))/(2) +/_ CBE = 90^(@) implies /_CBE = (90^(@) - (x^(@))/(2))`
`implies r = [ 90 - (x)/(2)]`.