Question

Let ABCD be a trapezium with parallel sides AB and CD such that the circle S with AB as its diameter touches CD. Further, the circle S passes through the midpoints of the diagonals AC and BD of the trapezium. The smallest angle of the trapezium is
A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(5)`
D. `(pi)/(6)`

Answer 1

Correct Answer - D

Join AN
`becauseangleANB=90^(@)`
`"In "triangleANB,`
`"cos "theta=(BQ)/(2r)`
`BN=2r "cos "theta`
`BD=2BN=4r "cos "theta`
`"In "triangleBQD`
`"sin "theta=(BQ)/(BD)=(r)/(4r" cos "theta)`
`"sin "2theta=(1)/(2)`
`theta=15^(@)`
`"Now similarly "alpha=15^(@)=theta " & "AC=4r "cos "alpha`
`therefore" Trapezium will be isosceles"`
`becauseangleADB=30^(@)`