ABCD is a kite having AB=AD and BC= CD prove that the quadrilaterals formed by joining the mid points of the sides is arectangle
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firstly draw a diagonal BD
and join the mid points of side ab and ad .let the mid points be e and f resp.
NOW ABD is triangle and by converse of midpoint theoram EF=1/2 BD AND EF//BD. Similarly in Δ BCD GH=1/2BD AND GH//BD (let g and h be midpoints of side CD AND BC resp.)
NOW, EF=GH AND EF//GH (by euclid's axiom no.1)
so,EFGH IS //grm
JOIN THE another diagonal ac and let the intersection be O
let the intersection of eh and ef with bd and ac be x and y resp.
now EX // yo and EY//XO SO, EXOH //grm
diagonals of kite bisect at 90° so ∠o=∠e=90°
∠feh=90°
so all angles are 90 hence prooved
and join the mid points of side ab and ad .let the mid points be e and f resp.
NOW ABD is triangle and by converse of midpoint theoram EF=1/2 BD AND EF//BD. Similarly in Δ BCD GH=1/2BD AND GH//BD (let g and h be midpoints of side CD AND BC resp.)
NOW, EF=GH AND EF//GH (by euclid's axiom no.1)
so,EFGH IS //grm
JOIN THE another diagonal ac and let the intersection be O
let the intersection of eh and ef with bd and ac be x and y resp.
now EX // yo and EY//XO SO, EXOH //grm
diagonals of kite bisect at 90° so ∠o=∠e=90°
∠feh=90°
so all angles are 90 hence prooved
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