Question

2. The diagonals AC and BD of a
parallelogram ABCD bisect each other at
O. A line segment XY through O has its
end-points on the opposite sides AB and
CD. Is XY also bisected at O ?​

Answer 1

$\underline{ Given: }$

$The \: diagonals \: AC \: and \: BD \: of \: a$

$parallelogram \: ABCD \: bisect \: each\:$

$other \: at \: O.$

$A \: line \: segment \:XY \: through \: O \: has$

$it's \: end - points \: on \: opposite \:sides$

$AB \: and \:CD$

$\underline { To \: Prove : }$

$\red{ XY \: bisected \: at \: O }$

$\underline{ Proof: }$

$In \: \triangle AOX \: and \: \triangle COY$

$\angle {AOX} = \angle {COY}$

$\blue { ( Vertically \: opposite \:angles )}$

$AO = OC$

$\blue { ( Diagonals \: bisects \: each \:other)}$

$\angle {OAX} = \angle {OCY}$

$\blue { ( Alternate \: angles ) }$

$\therefore \triangle AOX \cong \triangle COY$

$\green { ( A.S.A \: congruence \: rule )}$

$OX = OY \: ( C.P.C.T )$

$\green { O \: bisects \: XY }$

$Hence\: proved.$

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