The diagonals of a parallelogram ABCD intersect at O. A line through O intersects AB at X and DC at Y. Prove that OX = OY. [4 MARKS]

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Solution

Diagram : 1 Mark
Concept : 1 Mark
Proof : 2 Marks

Consider $Δ A X O$ and $Δ C Y O .$

$A O = C O$ [Diagonals bisect each other]

$∠ A O X = ∠ C O Y$ [Vertically opposite angles]

$∠ A X O = ∠ C Y O$ [Alternate angles]

$∴ Δ A X O ≅ Δ C Y O .$ [AAS congruency]

$O X = O Y$ [CPCT]

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The diagonals of a parallelogram ABCD intersect at O. A line through O intersects AB at X and DC at Y. Prove that OX = OY. [4 MARKS]

Diagram : 1 Mark Concept : 1 Mark Proof : 2 Marks Consider ΔAXO and ΔCYO. AO=CO [Diagonals bisect each other] ∠AOX=∠COY [Vertically opposite angles] ∠AXO=∠CYO [Alternate angles] ∴ΔAXO≅ΔCYO. [AAS congruency] OX=OY [CPCT]

Diagram : 1 Mark
Concept : 1 Mark
Proof : 2 Marks

Consider $Δ A X O$ and $Δ C Y O .$

$A O = C O$ [Diagonals bisect each other]

$∠ A O X = ∠ C O Y$ [Vertically opposite angles]

$∠ A X O = ∠ C Y O$ [Alternate angles]

$∴ Δ A X O ≅ Δ C Y O .$ [AAS congruency]

$O X = O Y$ [CPCT]