Question

ABCD is a square. EF is parallel to diagonal BD. AR is a line segment such that R is the mid-point of EF. Prove that :

i) BE=DF

ii) AR bisects angle BAD

iii) If AR is produced, it will pass through C.

For the diagram, please refer page no. 140, sum no. 20 of ICSE Foundation Mathematics by R.S. Aggarwal.

Answer 1

(i) according to thale's theorem ( theorem of proportionality )
in a triangle when a line is parallel to one of the side of triangle then the segment corresponding to one half sector is proportional to the segment of other half sector
so,
$\frac{be}{df} = \frac{ec}{cf}$
$be = df$