Question

8. In a parallelogram ABCD, the diagonals AC & BD intersect each other at O.

Through O, a line is drawn to intersect AD at X and BC at O. Through O, a line is drawn to intersect AD at X and BC at Y.

Show that OX = OY​

Answer 1

$\Large{\underline{\underline{\bold{☆Given:}}}}$

• ABCD is a Parallelogram.
• AC & BD are diagonals which intersect each other at O.
• A line which passes through point O intersect AD at X and BC at Y.

$\Large{\underline{\underline{\red{\bold{☆Required\;To\;Prove:}}}}}$

• XO = YO.

$\Large{\underline{\underline{\bold{\color{salmon}☆Proof:}}}}$

WKT

Diagonals of a Parallelogram intersect each other, So AO = CO & BO = DO.

In ∆AOX & ∆COY

• AO = CO [Proved Above]
• ∠AOX = ∠COY [Vertically Opposite Angles]

As, AB || CD & AC as transversal.

• ∠XAO = ∠YCO [Alternative Interior Angles]

∆AOX ≅ ∆COY [By ASA Axiom]

☞ OX = OY [By CPCT]

Hence Proved,

• $\boxed{\pink{\bold{OX\;=\;OY}}}$

_________________________

➸ Important Key Points:-

❂ραɾαℓℓεℓσɠɾαɱ:- A Quadrilateral in which two pair of opposite sides are parallel.

• → The opposite angles are equal and adjacent angles are supplementary.
• → The Diagonals bisect each other at a point.
• → Area of Parallelogram = Base×Height.

$\boxed{\tt{@MrSovereign}}$

Hope This Helps!!

Answer 2

Answer:-

ABCD is a Parallelogram.

AC & BD are diagonals which intersect each other at O.

A line which passes through point O intersect AD at X and BC at Y.

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XO = YO.

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☆Proof:

WKT

Diagonals of a Parallelogram intersect each other, So AO = CO & BO = DO.

In ∆AOX & ∆COY

AO = CO [Proved Above]

∠AOX = ∠COY [Vertically Opposite Angles]

As, AB || CD & AC as transversal.

∠XAO = ∠YCO [Alternative Interior Angles]

∆AOX ≅ ∆COY [By ASA Axiom]

☞ OX = OY [By CPCT]

Hence Proved,

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OX=OY

_________________________

➸ Important Key Points:-

❂ραɾαℓℓεℓσɠɾαɱ:- A

in which two pair of opposite sides are parallel.

→ The opposite angles are equal and adjacent angles are supplementary.

→ The Diagonals bisect each other at a point.

→ Area of Parallelogram = Base×Height.